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TheoremQA_standardized

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ABCD is a parallelogram. E is the midpoint, F is also a midpoint. Area of AFG = 10, Area of EGH = 3. What is Area CDH?

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7

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Find the arc length of y = (1/4)x^4 over the interval [1,2] using the Trapezoidal Rule T_5.

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3.958

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If a stock pays a $5 dividend this year, and the dividend has been growing 6% annually, what will be the stock’s intrinsic value, assuming a required rate of return of 12%?

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88.33

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

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3.70358145

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ABCD is a square. Inscribed Circle center is O. Find the the angle of ∠AMK. Return the numeric value.

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130.9

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Consider a $21 \times 17$ rectangular region. This region is to be tiled using tiles of the two types shown in ./mingyin/square1.png (The dotted lines divide the tiles into $1\times 1$ squares.) The tiles may be rotated and reflected, as long as their sides are parallel to the sides of the rectangular region. They must all fit within the region, and they must cover it completely without overlapping. What is the minimum number of tiles required to tile the region?

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99

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In how many ways can we color a loop of 5 vertices with 3 colors such that no two adjacent vertices have the same color?

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30

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Let f_1, ..., f_n be polynomials. Do they span the space P of all polynomials?

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False

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is the following function $f(t, y) = \frac{t^3+t^2y+ty+y^3}{t^3 + ty^2}$ scale invariant function

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True

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What is the smallest number of standard deviations from the mean that we must go if we want to ensure that we have at least 50% of the data of a distribution?

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1.4

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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?

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0.4295

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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?

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[3, 5]

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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]]?

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-252

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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?

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6

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A cascade of $n$ identical independent binary symmetric channels each with raw error probability $p$, and $0<p<1$. What is the capacity of the cascade when $n$ goes to infinity?

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0.0

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Let X_1, X_2 , X_3 be independent random variables taking values in the positive integers and having mass functions given by P(X_i=x)=(1-p_i)*p_i^{x-1} for x=1,2,... and i=1,2,3. Suppose p_1=1/2,p_2=1/4,p_3=1/8, what is the probability of X_1<X_2<X_3 (i.e. P(X_1<X_2<X_3))?

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0.00153609831

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

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True

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

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0.0

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Suppose a convex polygon has 26 faces and 39 edges. How many vertices does it have?

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15

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Is the Fourier transform of the signal x(t)=(1-e^{-|t|})[u(t+1)-u(t-1)] real?

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True

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compute the integral $\iint_{\Sigma} x^3 dy*dz +y^3 dz*dx+z^3 dx*dy$, where is the outward of the ellipsoid x^2+y^2+z^2/4=1. Round the answer to the thousands decimal.

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30.15928896

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A group of 10 people is split into 3 different committees of 3, 4, and 3 people, respectively. In how many ways can this be done?

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4200

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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?

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1.0

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Consider a forward contract on a 4-year bond with maturity 1 year. The current value of the bond is $1018.86, it has a face value of $1000 and a coupon rate of 10% per annum. A coupon has just been paid on the bond and further coupons will be paid after 6 months and after 1 year, just prior to delivery. Interest rates for 1 year out are flat at 8%. Compute the forward price of the bond.

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999.998976

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

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2

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Fig. Q2 shows a 1st-order noise shaper. The input is bounded by 0 v and 1 v. A constant 0.4 v input is fed into the noise shaper. The output is a periodic pattern sequence. What is the period of the sequence?

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5

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

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82.0

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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?

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176

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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)

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30

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An investor is looking to purchase a security for $100 with an initial margin of 50% (meaning the investor is using $50 of his money to purchase the security and borrowing the remaining $50 from a broker). In addition, the maintenance margin is 25%. At what price of the security will the investor receive a margin call?

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66.67

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Let h(x) = (x^{-1/2} + 2x)(7 - x^{-1}). What is h'(x) when x = 4?

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13.609

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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.

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1.32

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Let g(x) = 1 / (1 + x^{3/2}), what is g'(x) when x = 1?

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-0.375

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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.

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(b)

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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.

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False

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As shown in ./mingyin/integral1.png line $y=c$, $x=0$, and parabola $y=2x-3x^3$ splits the plane into the two shaded regions. Suppose two regions have the same areas. What is the value $c$?

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0.444444

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Which of the following matrices takes any vector $v$ and projects it onto the space spanned by the columns of $\Phi$? (a) $(\Phi^T\Phi)^{-1}$. (b) $\Phi(\Phi^T\Phi)^{-1}$. (c) $\Phi(\Phi^T\Phi)^{-1}\Phi^T$. (d) $\Phi^T(\Phi^T\Phi)^{-1}\Phi^T$.

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(c)

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A pure lead bar 10 cm long is maintained with one end at T &=300 K and the other at 310 K. The thermoelectric potential difference thus induced across the ends is 12.8 micro-volts. Find the thermoelectric power for lead in this temperature range in V/K. (Note: Q varies nonlinearly with temperature, but over this narrow temperature range, you may use a linear approximation.)

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1.28e-06

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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?

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7142

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Suppose f is an analytic function defined on $\{z \in C : IM(z) > 0\}$, the upper half plane. Given the information that f(f(z)) = z and f'(z) = 1/z^2 for every z. Find the most general possible expression of f(z). What is f(2)?

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-0.5

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How many trees are there on 5 labeled vertices?

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125

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Find the sum of $\sum_{n=1}^{\infty} \frac{2}{n^2 + 4n + 3}$

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0.8333

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Comet Halley moves in an elongated elliptical orbit around the sun (Fig. 13.20). Its distances from the sun at perihelion and aphelion are $8.75 \times 10^7 km$ and $5.26 \times 10^9 km$, respectively. The orbital period is X * 10^9 s. What is X?

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2.38

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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.

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

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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?

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0.251

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Find the largest integer for which (x+11)/(x+7) is an integer.

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-3

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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.)?

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26.0

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Suppose a fair coin is tossed 50 times. The bound on the probability that the number of heads will be greater than 35 or less than 15 can be found using Chebyshev's Inequality. What is the upper bound of the probability?

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0.125

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Given 3 Colors whose RGB representations are given as follows: Color 1: (0.5, 0.5, 0.5), Color 2: (0.4, 0.6, 0.5), Color 3: (0.3, 0.7, 0.5), Which Color does not carry chrominance (Color) Information? Answer with 1 or 2 or 3.

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1

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Let $A=\{n+\sum_{p=1}^{\infty} a_p 2^{-2p}: n \in \mathbf{Z}, a_p=0 or 1 \}$. What is the Lebesgue measure of A?

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0.0

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