Question stringlengths 28 1.07k | Answer stringlengths 1 54 | Answer_type stringclasses 6
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Let W(t) be the standard Brownian motion, and 0 < s < t. Find the conditional PDF of W(s = 1/2) given that W(t = 1) = 2. What are the mean and variance? Return the list of [mean, variance]. | [1.0, 0.25] | list of float | Not supported with pagination yet |
A scuba diver is wearing a head lamp and looking up at the surface of the water. If the minimum angle to the vertical resulting in total internal reflection is 25∘, what is the index of refraction of the water? $\theta_{air} = 1.00$. | 2.37 | float | Not supported with pagination yet |
For every positive real number $x$, let $g(x)=\lim _{r \rightarrow 0}((x+1)^{r+1}-x^{r+1})^{1/r}$. What is the limit of $g(x)/x$ as $x$ goes to infinity? | 2.7182818 | float | Not supported with pagination yet |
Find the orthogonal projection of 9e_1 onto the subspace of R^4 spanned by [2, 2, 1, 0] and [-2, 2, 0, 1]. | [8, 0, 2, -2] | list of integer | Not supported with pagination yet |
X rays scattered from rock salt (NaCl) are observed to have an intense maximum at an angle of 20° from the incident direction. Assuming n = 1 (from the intensity), what must be the Wavelength of the incident radiation in nm? | 0.098 | float | Not supported with pagination yet |
Suppose that there are two firms in the market facing no costs of production and a demand curve given by Q = 150 - P for their identical products. Suppose the two firms choose prices simultaneously as in the Bertrand model. Compute the prices in the nash equilibrium. | 0 | integer | Not supported with pagination yet |
In how many ways can 10 people be seated at 1 identical round tables? Each table must have at least 1 person seated. | 362880 | integer | Not supported with pagination yet |
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 distribution family $f_\theta(x)=\theta e^{-\theta x}$, $x \geq 0$? (a) $\theta$. (b) $\theta^2$. (c) $\theta^{-1}$. (d) $\theta^{-2}$. | (b) | option | Not supported with pagination yet |
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? | False | bool | Not supported with pagination yet |
Consider Convolutional Neural Network D2 which takes input images of size 32x32 with 1 colour channels. The first layer of D2 uses 4 filters of size 5x5, a stride of 2, and zero-padding of width 1. Consider CNN D2 which takes input images of size 32x32 with 1 colour channels. The first layer of D2 uses 4 filters of siz... | 900 | integer | Not supported with pagination yet |
How many ways are there to arrange 9 people in a line such that no one is standing in their correct position? | 133496 | integer | Not supported with pagination yet |
Ms. Fogg is planning an around-the-world trip on which she plans to spend $10,000. The utility from the trip is a function of how much she actually spends on it (Y), given by U(Y) = ln Y. If there is a 25 percent probability that Ms. Fogg will lose $1,000 of her cash on the trip, what is the trip’s expected utility? | 9.184 | float | Not supported with pagination yet |
In how many ways can 6 people be seated at 2 identical round tables? Each table must have at least 1 person seated. | 225 | integer | Not supported with pagination yet |
For a matrix A, is the function F(A) = det A from the linear space R^{3*3} to R a linear transformation? | False | bool | Not supported with pagination yet |
Are the vectors [1, 2], [2, 3], and [3, 4] linearly independent? | False | bool | Not supported with pagination yet |
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. | [51.2, 78.15, 125.0] | list of float | |
suppose $u=\arctan \frac{y}{x}$, what is numeric of $\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}$? | 0.0 | float | Not supported with pagination yet |
Use the Runge-Kutta method with $h=0.1$ to find approximate values of the solution of $(y-1)^2 * y' = 2x + 3$ with y(1) = 4. What is y(0)? | 3.46621207 | float | Not supported with pagination yet |
Define f: R o R by f(x) = (x^3) / (1 + x^2). Is f uniformly continuous on R? | True | bool | Not supported with pagination yet |
Evaluate $\int_c z^2 / (z - 5) dz$, where c is the circle that $|z| = 2$. | 0 | integer | Not supported with pagination yet |
Let a undirected graph G with edges E = {<0,1>,<1,3>,<0,3>,<3,4>,<0,4>,<1,2>,<2,5>,<2,7>,<2,6>,<6,7>,<6,10>,<5,8>,<10,9>,<5,10>,<6,8>,<7,8>,<6,9>,<7,10>,<8,10>,<9,11>,<9,12>,<9,13>,<13,12>,<13,11>,<11,14>}, which <A,B> represent Node A is connected to Node B. What is the shortest path from node 1 to node 14? Represent ... | [1, 2, 6, 9, 11, 14] | list of integer | Not supported with pagination yet |
The atomic mass of the 4He atom is 4.002603 u. Find the binding energy of the 4He nucleus in MeV. | 28.3 | float | Not supported with pagination yet |
Find the ratio of forward-bias to reverse-bias currents when the same voltage 1.5 V is applied in both forward and reverse. Assume room temperature 293 K. | -6e+25 | float | Not supported with pagination yet |
In how many ways can a group of 7 people be divided into 2 non-empty subsets? | 63 | integer | Not supported with pagination yet |
A debt of $25,000 is to be amortized over 7 years at 7% interest. What value of monthly payments will achieve this? | 4638.83 | float | Not supported with pagination yet |
Assume that half of the mass of a 62-kg person consists of protons. If the half-life of the proton is 10^33 years, calculate the number of proton decays per day from the body. | 3.5e-08 | float | Not supported with pagination yet |
A ship uses a sonar system to locate underwater objects. Find the wavelength of a 262-Hz wave in water. (Unit: m) | 5.65 | float | Not supported with pagination yet |
If polygon ABCDE ~ polygon PQRST, AB = BC = 8, AE = CD = 4, ED = 6, QR = QP, and RS = PT = 3, find the perimeter of polygon ABCDE. | 30 | integer | Not supported with pagination yet |
Are groups Z_4 * Z_2 and D_4 isomorphic? | False | bool | Not supported with pagination yet |
Are the vectors v_1 = [1,2,3], v_2 = [4,5,6], v_3 = [7,8,9] linearly independent? | False | bool | Not supported with pagination yet |
Let M be the inverse of the group element ((3, 5), (4, 6)) in Z_7. What is M[0][1]? | 6 | integer | Not supported with pagination yet |
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? | 140 | integer | Not supported with pagination yet |
How many distinct necklaces with 12 beads can be made with 10 beads of color R and 2 beads of color B, assuming rotations and reflections are considered equivalent? | 6 | integer | Not supported with pagination yet |
Find the arc length of the curve, where x=t, y=t^2 and z=2*t^3/3. | 7.333 | float | Not supported with pagination yet |
The planet Mercury travels around the Sun with a mean orbital radius of 5.8x10^10 m. The mass of the Sun is 1.99x10^30 kg. Use Newton's version of Kepler's third law to determine how long it takes Mercury to orbit the Sun. Give your answer in Earth days. | 88.3 | float | Not supported with pagination yet |
Consider $x(t)$ to be given as, $$ x(t)=10 \cos (20 \pi-\pi / 4)-5 \cos (50 \pi t) $$ What is minimum sampling rate (/Hz) such that $y(t)=x(t)$ ? | 50 | integer | Not supported with pagination yet |
Consider an arbitrage-free securities market model, in which the risk-free interest rate is constant. There are two nondividend-paying stocks whose price processes are:
$S_1(t)=S_1(0)e^{0.1t+0.2Z(t)}$
$S_2(t)=S_2(0)e^{0.125t+0.3Z(t)}$
where $Z(t)$ is a standard Brownian motion ant $t\ge0$. What is the continuously comp... | 0.02 | float | Not supported with pagination yet |
Use the Birge-Vieta method to find a real root correct to three decimals of the following equation: x^5 - x + 1 = 0, p=-1.5. | -1 | integer | Not supported with pagination yet |
Is the function of f(x) = sin(x) / |x| continuous everywhere? | False | bool | Not supported with pagination yet |
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? | True | bool | Not supported with pagination yet |
compute the integral \int_{\Gamma} \frac{x*dy-y*dx}{x^2+y^2}, where $\Gamma$ is any piecewise smooth, closed curve that encloses the origin but does not pass through it. | 6.2831852 | float | Not supported with pagination yet |
suppose the sequence a_n satisfies $lim_{n\rightarrow\infty}a_n\sum_{i=1}^n a_i^2=1$. What is the limit of 3n(a_n)^3? | 1.0 | float | Not supported with pagination yet |
given a finite group A, and a collection of permutations B. Then (a) there exists B such that A is isomorphic to B; (b) for any B, A is isomorphic to B; (c) A can never be isomorphic to B; (d) none of the above. Which option is correct? | (a) | option | Not supported with pagination yet |
what is the value of $\prod_{n=0}^{\infty}(1+(\frac{1}{2})^{2^n})$? | 2.0 | float | Not supported with pagination yet |
The diagonals of kite WXYZ intersect at P. If XP = 8, PZ = 8, WP = 6, and PY = 24, find ZY. | 25.3 | float | Not supported with pagination yet |
Astrophysical theory suggests that a burned-out star whose mass is at least three solar masses will collapse under its own gravity to form a black hole. If it does, the radius of its event horizon is X * 10^3 m, what is X? | 8.9 | float | Not supported with pagination yet |
Light travel from water n=1.33 to diamond n=2.42. If the angle of incidence was 13 degree, determine the angle of refraction. | 7.1 | float | Not supported with pagination yet |
An investor has utility function $U(x) = x^{1/4}$ for salary. He has a new job offer which pays $80,000 with a bonus. The bonus will be $0, $10000, $20000, $30000, $40000, $50000, or $60000, each with equal probability. What is the certainty equivalent value of this job offer? | 108610 | integer | Not supported with pagination yet |
Does \lim_{x \to 0} (cos(mx - 1)/(x^2) = -(m^2)/2 for m = 2? | True | bool | Not supported with pagination yet |
RS is the midsegment of trapezoid MNOP. If MN = 10x+3, RS=9x-1, and PO = 4x+7, what is the length of RS? | 26 | integer | Not supported with pagination yet |
The bandwidth of an analog signal is 4kHz. An A/D converter is used to convert the signal from analog to digital. What is the minimum sampling rate for eliminating the aliasing problem? (in kHz) | 8 | integer | Not supported with pagination yet |
Is the cumulative distribution function of the standard gaussian distribution $F(x)=1/\sqrt{2 \pi} \int_{-\infty}^x e^{-t^2/2} dt$ is log-concave? Return 1 for yes and 0 for no. | 1.0 | float | Not supported with pagination yet |
Compute the mean translational kinetic energy of a single ideal gas molecule in eV. | 0.038 | float | Not supported with pagination yet |
Let f_1, ..., f_n be polynomials. Do they span the space P of all polynomials? | False | bool | Not supported with pagination yet |
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? | 78.3526 | float | Not supported with pagination yet |
What is the value of the integral $\int_0^{\pi/2} 1/(1+(tan(x))^{\sqrt{2}}) dx$? | 0.78539815 | float | Not supported with pagination yet |
How many labeled graphs with a score of (6, 2, 2, 2, 2, 2, 2) are there? | 15 | integer | Not supported with pagination yet |
A hydraulic press contains $0.25 m^3$ (250 L) of oil. Find the decrease in the volume of the oil when it is subjected to a pressure increase $\Delta p=1.6 \times 10^7 Pa$ (about 160 atm or 2300 psi). The bulk modulus of the oil is $B=5.0 \times 10^9 Pa$ (about $5.0 \times 10^4 atm$) and its compressibility is $k=1 / B... | -0.8 | float | Not supported with pagination yet |
Two sets of points are linearly separable if and only if their convex hulls are disjoint. True or False? | True | bool | Not supported with pagination yet |
A Chord based distributed hash table (DHT) with 25 address space is used in a peer- to-peer file sharing network. There are currently 5 active peers in the network with node ID N3, N8, N15, N19 and N30. Show all the target key (in ascending order, ignore the node's identifier itself) for N3. | [4, 5, 7, 11, 19] | list of integer | Not supported with pagination yet |
What is $\lim _{r \rightarrow \infty} (\int_0^{\pi/2} x^r sin(x) dx)/(r\int_0^{\pi/2} x^r cos(x) dx)$? | 0.63662 | float | Not supported with pagination yet |
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 | 2.37 | float | Not supported with pagination yet |
Is the Fourier transform of the signal $x_1(t)=\left\{\begin{array}{cc}\sin \omega_0 t, & -\frac{2 \pi}{\omega_0} \leq t \leq \frac{2 \pi}{\omega_0} \\ 0, & \text { otherwise }\end{array}\right.$ imaginary? | True | bool | Not supported with pagination yet |
suppose $-\pi<x<\pi$. what is the value of $(\sum_{n=1}^{\infty}(-1)^{n-1} \frac{cos(nx)}{n})/log(2cos(x/2))$? Rounding it to the hundredths place and return the value. | 1.0 | float | Not supported with pagination yet |
Find the curvature for r(t) = 5cos(t)i + 4sin(t)j + 3tk, t=4\pi/3. | 0.16 | float | Not supported with pagination yet |
What is 3^(3^(3^(...))) mod 100? There are 2012 3's in the expression. | 87 | integer | Not supported with pagination yet |
Compute the real integral $I=\int_{-\infty}^{\infty} 1/(x^2 + 1)^2 dx$. | 1.57 | float | Not supported with pagination yet |
How many different 6-letter arrangements can be made from the letters in the word BANANA? | 60 | integer | Not supported with pagination yet |
Consider a 26-key typewriter. Suppose that pushing a key results in printing that letter or the next (with equal probability). Thus A results in A or B, ..., Z results in Z or A. What is the capacity of this channel in bits? | 3.7 | float | Not supported with pagination yet |
Suppose H is a Banach space, and {x_n}\in H, x\in H. Then x_n weakly converges to x is equivalent to: ||x_n|| is bounded; for a dense set M* in H*, it holds \lim_{n\rightarrow\infty} f(x_n)=f(x) for all f\in M*. Is this correct? Answer 1 for yes and 0 for no. | 1.0 | float | Not supported with pagination yet |
Given 2 colors whose HSI representations are given as follows: which color looks closer to blue? (a) Color 1: $(\pi, 0.3,0.5)$, (b) Color 2: $(0.5 \pi, 0.8,0.3)$ | (a) | option | Not supported with pagination yet |
Fig. Q7a shows the amplitude spectrum of a real-value discrete time signal x[n]. Determine the period of signal x[n] (in samples). | 8 | integer | |
Consider a 900 Kbytes file stored in a web server. Client A sends a request to the server to retrieve the file from a remote location. There are 3 links (2 intermediate nodes) between server and client and each has a transmission rate of 10Mbps. Given that the segment size is 15 Kbytes, the round trip time (RTT) betwee... | 918 | integer | Not supported with pagination yet |
Suppose a convex polygon has 26 faces and 39 edges. How many vertices does it have? | 15 | integer | Not supported with pagination yet |
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. | [12, 8, 5] | list of integer | |
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. | [6, 7, 0, 6, 6, 4, 3, 4, 3, 1, 1] | list of integer | |
The earth and sun are 8.3 light-minutes apart. Ignore their relative motion for this problem and assume they live in a single inertial frame, the Earth-Sun frame. Events A and B occur at t = 0 on the earth and at 2 minutes on the sun respectively. Find the time difference in minutes between the events according to an o... | 14 | integer | Not supported with pagination yet |
suppose I=[0,1]\times[0,1], where exp is the exponential function. What is the numeric of the double integral of the function f(x,y)=x*y^3 exp^{x^2+y^2} over I? | 0.4295 | float | Not supported with pagination yet |
Given that $V_A = V_B$, determine the value of $C_2$ (in μF) in the following circuit in the figure. | 0.103 | float | |
Carl the clothier owns a large garment factory on an isolated island. Carl's factory is the only source of employment for most of the islanders, and thus Carl acts as a monopsonist. The supply curve for garment workers is given by l = 80w, where l is the number of workers hired and w is their hourly wage. Assume also t... | 200 | integer | Not supported with pagination yet |
A glass contains 0.25 kg of Omni-Cola (mostly water) initially at 25°C. How much ice, initially at -20°C must you add to obtain a final temperature of 0°C with all the ice melted? Neglect the heat capacity of the glass. (Unit: g) | 70 | integer | Not supported with pagination yet |
The diagonals of rhombus QRST intersect at P. If m∠QTS = 76, find m∠TSP. | 52 | integer | Not supported with pagination yet |
What is the determinant of the matrix A = [[1, 0, 0, 0, 0, 0], [2, 7, 0, 0, 0, 0], [3, 8, 6, 0, 0, 0], [4, 9, 5, 2, 1, 4], [5, 8, 4, 0, 2, 5], [6, 7, 3, 0, 3, 6]]? | -252 | integer | Not supported with pagination yet |
Maximize the entropy $H(X)$ of a non-negative integer-valued random variable $X$, taking values from 0 to infinity, subject to the constraint $E(X)=1$. Use base 2 logarithm to evaluate $H(X)$. | 2.0 | float | Not supported with pagination yet |
The 4 8x8 images shown below are encoded with JPEG coding. Based on their expected DCT (Discrete Cosine Transform) coefficients, Which image has the most non-zero AC coefficients? (a): Image A, (b): Image B, (c): Image C, (d): Image D. | (b) | option | |
Your firm is trying to decide whether to buy an e-commerce software company. The company has $100,000 in total capital assets: $60,000 in equity and $40,000 in debt. The cost of the company’s equity is 10%, while the cost of the company's debt is 5%. The corporate tax rate is 21%. What is the WACC of the company? | 0.0758 | float | Not supported with pagination yet |
is the following function $f(t, y) = \frac{t^3+t^2y+ty+y^3}{t^3 + ty^2}$ scale invariant function | True | bool | Not supported with pagination yet |
In triangle ACD, B is located on the side AC, and E is located on the side AD. If AB = 3, AC = 5, CD = 3.5, ED = 3, and EB ∥ DC, what is the length of AD? | 7.5 | float | Not supported with pagination yet |
An ultrasonic transducer used for medical diagnosis oscillates at 6.7 Mhz.How long does each oscillation take, and what is the angular frequency? (Unit: 10^7 rad/s) | 4.2 | float | Not supported with pagination yet |
How many pairs of (a, b) can we substitute for a and b in 30a0b03 so that the resulting integer is divisible by 13? | 3 | integer | Not supported with pagination yet |
A gun is designed that can launch a projectile of mass 10 kg at a speed of 200 m/s. The gun is placed close to a straight, horizontal railway line and aligned such that the projectile will land further down the line. A small rail car of mass 200 kg and travelling at a speed of 100 m/s passes the gun just as it is fired... | 60.0 | float | Not supported with pagination yet |
Let x \in R with 0 < x < 1 and n \in N. Is (1 - x)^n >= 1/(1+nx)? | False | bool | Not supported with pagination yet |
Consider a strategy of the form $(\gamma, 0, 0)$ for the investment wheel. Show that the overall factor multiplying your money after $n$ steps is likely to be $(1+2\gamma)^{n/2}(1-\gamma)^{n/2}$. Find the value of $\gamma$ that maximizes this factor. | 0.25 | float | Not supported with pagination yet |
What is the Fisher information for the distribution family $f_\theta(x)=\theta e^{-\theta x}$, $x \geq 0$? (a) $\theta$. (b) $\theta^2$. (c) $\theta^{-1}$. (d) $\theta^{-2}$. Which option is correct? | (d) | option | Not supported with pagination yet |
A teacher wants to invest $30,000 into an account that compounds annually. The interest rate at this bank is 1.8%. How much money will be in the account after 6 years? | 33389.35 | float | Not supported with pagination yet |
Compute the mean molecular speed v in the heavy gas radon (Rn) in m/s | 167.0 | float | Not supported with pagination yet |
In a set of 20 positive integers, at least how many pairs of numbers have a difference that is a multiple of 10? | 10 | integer | Not supported with pagination yet |
A symmetric random walk on the three-dimensional cubic lattice Z^3 is transient or persistent? Return 1 for persistent and 0 for transient. | 0.0 | float | Not supported with pagination yet |
Let C[0,1] be all the continuous function on in the interval [0,1]. For the integral equation $x(t)-\lambda \int_0^1 e^{t-s} x(s) ds=y(t)$, where $y(t)\in C[0,1]$ is a given function. \lambda is a constant and |\lambda|<1. Then there exists a unique solution x(t)\in C[0,1]. This conclusion can be proved by: 1. Implicit... | 3.0 | float | Not supported with pagination yet |
A neutron at rest decays (breaks up) to a proton and an electron. Energy is released in the decay and appears as kinetic energy of the proton and electron. The mass of a proton is 1836 times the mass of an electron. What fraction of the total energy released goes into the kinetic energy of the proton? | 0.000544 | float | Not supported with pagination yet |
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