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Two argon atoms form the molecule $Ar_2$ as a result of a van der Waals interaction with $U_0 = 1.68 \times 10 ^ {-21}$ J and $R_0 = 3.82 \times 10 ^ {-10}$ m. Find the frequency of small oscillations of one Ar atom about its equilibrium position. (Unit: 10^11 Hz)

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5.63

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One is given a communication channel with transition probabilities $p(y|x)$ and channel capacity $C=max_{p(x)}I(X;Y)$. If we preprocesses the output by forming $Y=g(Y)$ the capacity will not improve. True or False?

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True

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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?

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True

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\lim_{x \to 1}(1/(x - 1) - c/(x^3 - 1)) exists. What is the value of c?

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3

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For the function $f(x)=|x|−1$ defined on $[-1,1]$. Does it meet the criteria of Rolle's Theorem? Answer true or false.

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False

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

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5.667

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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, and a remainder of 4 when divided by 5.

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59

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

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False

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In how many ways can a group of 6 people be divided into 2 teams? Notice that members in each team are ordered.

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1800

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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.

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0.25

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

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0.0

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for a given function f(x)=x^2*sin(x). Is there a value $x$ between 10pi and 11pi such that $f'(x) = 0$?

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True

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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.).

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417

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suppose f is differentiable in [0,+\infty) and f(0)=0. When x>=0, |f'(x)|<=|f(x)| where f' stands for the derivative of f. What is f(2687) and f(35)? answer the two values in a list

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[0, 0]

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An aluminum cylinder 10 cm long, with a cross-sectional area of 20 $cm^2$ is used as a spacer between two steel walls. At 17.2°C it just slips between the walls. Calculate the stress in the cylinder and the total force it exerts on each wall when it warms to 22.3°C assuming that the walls are perfectly rigid and a constant distance apart. (Unit: 10^4 N)

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-1.7

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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)

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1.4

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A surveyor uses a steel measuring tape that is exactly 50.000 m long at a temperature of 20°C. The markings on the tape are calibrated for this temperature. When it is 35°C, the surveyor uses the tape to measure a distance. The value that she reads off the tape is 35.794 m. What is the actual distance? (Unit: m)

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35.8

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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?

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(c)

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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.

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344

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Calculate the de Broglie Wavelength, in nm, of an electron with kinetic energy 50 eV.

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0.17

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what is the value of \int_a^b \frac{dx}{\sqrt{(x-a)(b-x)}}? Round the answer to the thousands decimal.

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3.1415926

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If z = \frac{1 + e^{-2x}}{x + tan(12x)}, what's the derivative of $\frac{\partial z}{\partial x}$ at $x = 1$.

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-153.59

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The root of the equation x = (1 / 2) + sin x by using the iteration method: x_{k+1} = 1/2 + sin(x_k), x_0 = 1 correct to o six decimals is x = 1.497300. Determine the number of iteration steps required to reach the root by linear iteration. If the Aitken ∆2-process is used after three approximations are available, how many iterations are required?

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3

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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.

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1.9221

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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.

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1.0

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Determine the period of the following signal, $$ x_1(t)=\cos (3 \pi t)-4 \cos (5 \pi t-0.5 \pi) $$

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2

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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. The dimensions of the resulting activation map for each filter in this first layer will be k x k. What is the value of k?

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15

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

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[-1, 6]

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Light of wavelength 400 nm is incident upon lithium (phi = 2.93 eV). Calculate the photon energy in eV.

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3.1

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The distortion rate function $D(R)=\min_{p(\hat{x}|x):I(X;\hat{X})\leq R} E(d(X,\hat{X}))$ is nonincreasing. True or False?

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True

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Use divergence therem to evaluate $\iint_S \vec{F} \cdot d \vec{S}$ where $\vec{F} = yx^2 \vec{i} + (xy^2 - 3z^4)\vec{j} + (x^3+y^3)\vec{k}$ and the surface $S$ consists of the sphere of radius 4 with $z \le 0$ and $y \le 0$. Note all three surfaces of this solid are included in $S$.

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0.0

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Let a undirected graph G with edges E = {<0,2>, <2,4>, <3,4>, <1,4>}, which <A,B> represent Node A is connected to Node B. What is the minimum vertex cover of G if 0 is one of vertex cover? Represent the vertex cover in a list of ascending order.

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[0, 4]

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For how many positive integral values of x ≤ 100 is 3^x − x^2 divisible by 5?

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20

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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.

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19

output

If at the beginning of each month a deposit of $500 is made in an account that pays 8% compounded monthly, what will the final amount be after five years?

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36983.35

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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.

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False

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For the signal f(t)=3sin(200πt)+ 6sin(400πt) + sin(500πt), determine the minimum sampling requency (in πHz) satisfying the Nyquist criterion.

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500

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

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False

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Let V be the space spanned by functions cos(2x) and sin(2x). Find the determinant of the linear transformation D(f) = f' from V to V.

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4

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ABCD is a parallelogram such that AB is parallel to DC and DA parallel to CB. The length of side AB is 20 cm. E is a point between A and B such that the length of AE is 3 cm. F is a point between points D and C. Find the length of DF in cm such that the segment EF divide the parallelogram in two regions with equal areas.

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17

output

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.

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8

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Is the differential equation $2tyy' + 2t + ty^2 = 0$ the total derivative of the potential function $\phi(t, y) = t^2 + ty^2$?

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False

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In the figure, what is the magnitude of the potential difference across the $20 \Omega$ resistor? Answer in unit of W (3 sig.fig.).

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7.76

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Suppose the demand curve for oPads is given by $p=\frac{500-x}{10}, What is the elasticity value of this demand function.

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-1.5

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We are interested in the capacity of photographic film. The film consists of silver iodide crystals, Poisson distributed, with a density of 100 particles per unit area. The film is illuminated without knowledge of the position of the silver iodide particles. It is then developed and the receiver sees only the silver iodide particles that have been illuminated. It is assumed that light incident on a cell exposes the grain if it is there and otherwise results in a blank response. Silver iodide particles that are not illuminated and vacant portions of the film remain blank. We make the following assumptions: We grid the film very finely into cells of area $dA$. It is assumed that there is at most one silver iodide particle per cell and that no silver iodide particle is intersected by the cell boundaries. Thus, the film can be considered to be a large number of parallel binary asymmetric channels with crossover probability $1 - 100dA$. What is the capacity of a 0.1 unit area film?

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10.0

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Use Green's Theorem to evaluate $\oint_{C} xy dx + x^2y^3dy$ where $C$ is the triangle with vertices (0,0), (1,0), (1,2) with positive orientation

instruction

0.6667

output

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

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4200

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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.

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937

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

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1.0

output

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?

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True

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