instruction_seed stringlengths 56 2.52k | source stringclasses 1
value | embeddings listlengths 1.02k 1.02k | mean_positive_score float64 0.77 0.86 | mean_negative_score float64 0.76 0.85 | difference_score float64 0 0.03 |
|---|---|---|---|---|---|
Suppose that a factory produces two kinds of products, A and B, each requiring two stages of production. The production of product A in the first stage and the production of product B in the second stage are subject to stochastic disturbances. The cost of producing a unit of product A in the first stage is $5, the cost... | tiger_lab_math | [
0.017635256052017212,
0.011797916144132614,
-0.06691553443670273,
-0.01226781401783228,
0.037401508539915085,
0.008435491472482681,
-0.049235787242650986,
0.09737981110811234,
0.0354543998837471,
0.004139855969697237,
0.011119605042040348,
0.028433380648493767,
-0.017512107267975807,
-0.02... | 0.800431 | 0.781158 | 0.019273 |
A company has to deliver goods to 5 different cities. The distances between the cities are as follows: city 1 to city 2 is 100 km, city 1 to city 3 is 250 km, city 1 to city 4 is 200 km, city 1 to city 5 is 150 km, city 2 to city 3 is 150 km, city 2 to city 4 is 100 km, city 2 to city 5 is 200 km, city 3 to city 4 is 1... | tiger_lab_math | [
0.0027657870668917894,
0.02535763382911682,
-0.02832716330885887,
-0.04056596755981445,
0.03200354799628258,
-0.009421246126294136,
-0.04160143435001373,
0.04621592164039612,
0.026633689180016518,
-0.010983377695083618,
0.004542023409157991,
0.03968977928161621,
-0.00820688996464014,
-0.01... | 0.806717 | 0.801349 | 0.005368 |
Let X be the unit sphere in three-dimensional space, and let p: X → S² be the standard projection map. Determine the fundamental group of S² \ {p((1, 0, 0))}. | tiger_lab_math | [
0.012197651900351048,
0.01915614679455757,
-0.04335774853825569,
-0.044193148612976074,
0.03752117604017258,
-0.03771413490176201,
-0.014211457222700119,
0.041103314608335495,
0.040681783109903336,
0.0005039267707616091,
0.018848810344934464,
0.031243890523910522,
-0.02236475609242916,
-0.... | 0.818737 | 0.80155 | 0.017187 |
One possible question for a math student to solve regarding Conformal mapping and its applications in Complex Analysis is:
Find a conformal map that transforms the unit disk $D(0,1)$ onto the domain bounded by the real axis and the curve $y=|x|^{\frac{3}{2}}$ in the complex plane.
Hint: Try to first find a map that t... | tiger_lab_math | [
0.024613775312900543,
0.042200878262519836,
-0.04211084544658661,
-0.04630056768655777,
0.018794184550642967,
0.00021872128127142787,
0.0017329785041511059,
0.03652007505297661,
0.03367798030376434,
-0.005936238914728165,
0.014087370596826077,
0.02582973800599575,
-0.015354963950812817,
-0... | 0.818211 | 0.799689 | 0.018522 |
In convex quadrilateral $KLMN$ side $\overline{MN}$ is perpendicular to diagonal $\overline{KM}$, side $\overline{KL}$ is perpendicular to diagonal $\overline{LN}$, $MN = 65$, and $KL = 28$. The line through $L$ perpendicular to side $\overline{KN}$ intersects diagonal $\overline{KM}$ at $O$ with $KO = 8$. Find $MO$.
| tiger_lab_math | [
0.0096824262291193,
0.019952906295657158,
-0.04077162593603134,
-0.05199803411960602,
0.03437946364283562,
0.004565275274217129,
-0.021022219210863113,
0.05413056164979935,
0.029785409569740295,
0.005263048689812422,
0.031014151871204376,
0.03251371160149574,
-0.045499563217163086,
-0.0095... | 0.832508 | 0.819131 | 0.013377 |
In $\triangle XYZ$, we have $\angle X = 90^\circ$ and $\tan Z = 3$. What is $\cos Z$? Let's write a Python program to solve it. | tiger_lab_math | [
0.018244080245494843,
0.023772763088345528,
-0.036091867834329605,
-0.03788761422038078,
0.027804017066955566,
-0.011647366918623447,
-0.015237011015415192,
0.02601831778883934,
0.0409892201423645,
-0.02240421436727047,
0.030453398823738098,
0.02442762814462185,
-0.03381749615073204,
0.001... | 0.829046 | 0.827059 | 0.001987 |
Consider the dynamical system given by the differential equation:
dx/dt = 2x - y
dy/dt = x + 2y
The Poincaré map for this system is given by:
P(x,y) = (x(T), y(T))
where T is the period of the system obtained by solving the equations. Determine the Poincaré map for the initial condition x(0) = 1 and y(... | tiger_lab_math | [
0.02617594785988331,
0.012472206726670265,
-0.037179213017225266,
-0.04691280797123909,
0.02744125947356224,
-0.019920695573091507,
-0.02008376643061638,
0.09562356024980545,
0.03427314758300781,
0.001209427136927843,
0.006000074092298746,
0.0307461004704237,
-0.028902405872941017,
-0.0307... | 0.808225 | 0.796948 | 0.011277 |
In a Public Goods Game, there are 4 players. Each player can choose to contribute either 0 or 2 units of money towards a public fund, where the total amount of money is multiplied by 1.5 and evenly distributed among all players. If all players contribute 2 units of money, each player will get 6 units of money. However,... | tiger_lab_math | [
0.004538710694760084,
0.00616537407040596,
-0.03476312384009361,
-0.020294738933444023,
0.03339502587914467,
-0.020989399403333664,
-0.038149282336235046,
0.030890192836523056,
0.026912054046988487,
-0.026485851034522057,
0.014819974079728127,
0.023771334439516068,
-0.02504822239279747,
-0... | 0.809438 | 0.803314 | 0.006124 |
In an isosceles triangle, one of the angles opposite an equal side is $40^{\circ}$. How many degrees are in the measure of the triangle's largest angle? [asy] draw((0,0)--(6,0)--(3,2)--(0,0)); label("$\backslash$",(1.5,1)); label("{/}",(4.5,1));
label("$40^{\circ}$",(.5,0),dir(45));
[/asy] | tiger_lab_math | [
0.02003352716565132,
0.028460318222641945,
-0.019589699804782867,
-0.04673352465033531,
0.032312363386154175,
-0.011941988952457905,
-0.01924341730773449,
0.0409514456987381,
0.022925730794668198,
-0.0036514049861580133,
0.03944532200694084,
0.029301997274160385,
-0.02370862103998661,
-0.0... | 0.83169 | 0.828664 | 0.003026 |
Let $\mathbf{a}$ and $\mathbf{b}$ be vectors such that $\|\mathbf{a}\| = 2,$ $\|\mathbf{b}\| = 5,$ and $\|\mathbf{a} \times \mathbf{b}\| = 8.$ Find $|\mathbf{a} \cdot \mathbf{b}|.$ Please write a program to solve it | tiger_lab_math | [
0.0401364341378212,
0.04498773440718651,
-0.05260709673166275,
-0.04041794687509537,
0.028432223945856094,
0.0028273563366383314,
-0.033505119383335114,
0.07286042720079422,
0.02879028208553791,
-0.020543843507766724,
0.00547042116522789,
0.03271873667836189,
-0.02924484759569168,
-0.03640... | 0.837399 | 0.832167 | 0.005232 |
In right triangle $JKL$, angle $J$ measures 60 degrees and angle $K$ measures 30 degrees. When drawn, the angle bisectors of angles $J$ and $K$ intersect at a point $M$. What is the measure of obtuse angle $JMK$?
[asy]
import geometry;
import olympiad;
unitsize(0.8inch);
dotfactor = 3;
defaultpen(linewidth(1pt)+fontsi... | tiger_lab_math | [
0.021736182272434235,
0.009543320164084435,
-0.041159432381391525,
-0.0517866425216198,
0.03324175998568535,
-0.005468354560434818,
-0.0255355853587389,
0.0444759801030159,
0.04604789614677429,
0.0023470206651836634,
0.035051003098487854,
0.026903945952653885,
-0.03565225005149841,
-0.0232... | 0.819431 | 0.808522 | 0.010909 |
keats library purchases a number of new books , all in the category of biography , and the library does not acquire any other books . with the addition of the new biographies , the biography collection of the library amounts to 55 % of the total number of books in the library . if prior to the purchase , only 20 % of t... | tiger_lab_math | [
0.02255207858979702,
0.015454737469553947,
-0.05014102905988693,
-0.02447003684937954,
0.03983651101589203,
-0.0443388931453228,
-0.022987257689237595,
0.04292209446430206,
0.054768357425928116,
-0.0060614305548369884,
0.011993590742349625,
0.0010167460422962904,
-0.04938226938247681,
0.00... | 0.806762 | 0.802822 | 0.00394 |
Two friends, Alex and Taylor, want to go out for the evening but cannot agree on where to go. Alex prefers to go to the movies while Taylor prefers to go to a concert. They can either go to the movies or the concert, but only to one location for the evening. The payoff matrix for their choices is:
| | ... | tiger_lab_math | [
0.0523294098675251,
0.000359829660737887,
-0.0497652143239975,
-0.011012353003025055,
0.02793239802122116,
-0.01690315268933773,
-0.029537582769989967,
0.07371817529201508,
0.035993412137031555,
-0.023009339347481728,
0.019606001675128937,
0.02724473364651203,
-0.04431076720356941,
-0.0350... | 0.817208 | 0.809671 | 0.007537 |
Consider the following partial differential equation for image processing:
∂u/∂t = k(∂²u/∂x² + ∂²u/∂y²)
where u(x,y,t) represents a 2D image at time t, k is a constant, and ∂²u/∂x² and ∂²u/∂y² represent the second partial derivatives of u with respect to x and y, respectively.
For an image with the initial conditi... | tiger_lab_math | [
-0.025637511163949966,
0.027088157832622528,
-0.05031370371580124,
-0.028017563745379448,
0.04982754588127136,
0.0035662308800965548,
-0.021222921088337898,
0.08819419890642166,
0.05165904387831688,
-0.008701568469405174,
0.03214073181152344,
0.011981573887169361,
-0.011146587319672108,
-0... | 0.801455 | 0.785379 | 0.016076 |
Consider the algebraic curve given by the equation $y^2 = x^3 - x^2$.
a) Use the implicit function theorem to find the equation of the tangent line to the curve at the point $(1,0)$.
b) Determine whether the curve has a singular point at the origin $(0,0)$. | tiger_lab_math | [
0.008502531796693802,
0.030808301642537117,
-0.04624544456601143,
-0.029546961188316345,
0.03430026397109032,
-0.02627991884946823,
-0.004003289621323347,
0.07577703893184662,
0.031088868156075478,
-0.005799789447337389,
-0.010032557882368565,
0.01713656634092331,
-0.02522316947579384,
-0.... | 0.824856 | 0.807111 | 0.017745 |
Let $\omega = x dy \wedge dz + y dz \wedge dx + z dx \wedge dy$ be a differential $2$-form on $\mathbb{R}^3$. Calculate its exterior derivative. | tiger_lab_math | [
-0.00934692658483982,
0.035007450729608536,
-0.05217215046286583,
-0.054222069680690765,
0.018593525514006615,
0.005243351217359304,
-0.03218040242791176,
0.046181656420230865,
0.033783797174692154,
-0.0066363876685500145,
0.009905786253511906,
0.03325219079852104,
-0.05537215620279312,
-0... | 0.821167 | 0.818715 | 0.002452 |
Let A = {1, 2} and B = {3, 4}, find:
1. A x B
2. A' x B'
3. (A x B)'
4. A x B'
5. A' x B
Verify if the Cartesian product of two sets is distributive over complement or not. | tiger_lab_math | [
0.04723191633820534,
0.025599565356969833,
-0.02589195966720581,
-0.012260040268301964,
0.031183648854494095,
-0.03009650483727455,
-0.01159296091645956,
0.06704634428024292,
0.05331509932875633,
-0.017673728987574577,
0.01293269544839859,
0.03294592723250389,
-0.025305666029453278,
-0.019... | 0.839514 | 0.828155 | 0.011359 |
A Gosper curve is constructed by starting with an equilateral triangle and repeatedly replacing each line segment with a specific pattern. The length of the initial line segment is 1 unit. If we continue this pattern for 4 iterations, what is the total length of the curve? | tiger_lab_math | [
-0.0007941584917716682,
-0.004556140396744013,
-0.019590280950069427,
-0.05437115207314491,
0.03604775294661522,
-0.026337333023548126,
-0.04398446902632713,
0.06911800056695938,
0.03972318768501282,
-0.02597493678331375,
0.019935419782996178,
0.03228411450982094,
-0.021960794925689697,
-0... | 0.819456 | 0.818396 | 0.001059 |
A manufacturing plant produces two types of products, A and B. The profit from each unit of product A is $10 and from each unit of product B is $15. The plant can produce a maximum of 500 units per day, and the production of product B cannot exceed the production of product A. The plant must produce at least 100 units ... | tiger_lab_math | [
0.020824585109949112,
0.005924507975578308,
-0.06680716574192047,
0.0048440308310091496,
0.012589216232299805,
-0.03294660151004791,
-0.0469403900206089,
0.09359674900770187,
0.029009545221924782,
-0.008975628763437271,
-0.0050475639291107655,
0.02513326331973076,
-0.010632912628352642,
-0... | 0.805 | 0.79035 | 0.01465 |
Consider the 2-sphere $S^2$ with an equatorial circle $C$ of radius 1. Let $D$ be the closed disk of radius 1 centered at a point on $C$, and let $D'$ be the interior of $D$. Find the smallest non-negative integer $n$ such that there is a continuous map from $D$ to $S^2 - C$ that misses at least $n$ points of $S^2 - C$... | tiger_lab_math | [
0.011154421605169773,
0.032392315566539764,
-0.03291977941989899,
-0.03406260162591934,
0.03141600638628006,
-0.008325156755745411,
-0.018287576735019684,
0.04295263811945915,
0.023025866597890854,
-0.014051566831767559,
0.019716929644346237,
0.032550208270549774,
-0.012851914390921593,
0.... | 0.856502 | 0.83308 | 0.023422 |
Suppose you have two sets of data: Set A and Set B. Set A has 10 data points with a mean of 25 and a standard deviation of 5, while Set B has 12 data points with a mean of 20 and a standard deviation of 3. Using discriminant analysis, determine the boundary (or decision) value that separates the two sets with a classif... | tiger_lab_math | [
0.007449025753885508,
-0.009592675603926182,
-0.05939598008990288,
-0.033094294369220734,
0.026326455175876617,
-0.046207159757614136,
-0.03425220400094986,
0.061457112431526184,
0.05798565596342087,
-0.019725289195775986,
0.008142407983541489,
0.024170273914933205,
-0.035940222442150116,
... | 0.799697 | 0.796134 | 0.003562 |
Suppose a principal hires an agent to carry out a project. The project's success is dependent on the effort put in by the agent. The principal can monitor the project's progress but cannot directly observe the effort put in by the agent. Let the project's success be represented by a random variable S which takes a valu... | tiger_lab_math | [
-0.007495403289794922,
0.015501345507800579,
-0.042786866426467896,
-0.026807544752955437,
0.034289903938770294,
-0.018158989027142525,
-0.02657858282327652,
0.04937826842069626,
0.01981476880609989,
-0.001101500354707241,
0.030149733647704124,
0.02632679045200348,
-0.03664159029722214,
-0... | 0.820928 | 0.808564 | 0.012364 |
Consider a game in which two players, A and B, simultaneously choose either "heads" or "tails" by placing a penny on the table. If the pennies match, player A wins $1 from player B. If the pennies do not match, player B wins $1 from player A. Suppose player A has a probability of choosing "heads" equal to 0.6, and play... | tiger_lab_math | [
0.006699707359075546,
0.017867734655737877,
-0.050763506442308426,
-0.020564261823892593,
0.037278685718774796,
-0.004376049153506756,
-0.010728905908763409,
0.046601518988609314,
0.048416540026664734,
-0.02956334687769413,
0.026897020637989044,
0.030542198568582535,
-0.029573090374469757,
... | 0.819088 | 0.806262 | 0.012826 |
A delivery truck has to deliver packages to 5 different locations in a city. Each location has a specific distance from the starting point of the truck, as shown in the table below:
| Location | Distance (km) |
|----------|--------------|
| A | 6 |
| B | 12 |
| C | 8 ... | tiger_lab_math | [
0.0021012555807828903,
0.01690829172730446,
-0.032340023666620255,
-0.03522942215204239,
0.028412995859980583,
-0.02188878133893013,
-0.04836445674300194,
0.03286370635032654,
0.04359325394034386,
-0.019792499020695686,
0.011758276261389256,
0.031270623207092285,
-0.01951764151453972,
0.00... | 0.820651 | 0.815821 | 0.004831 |
A store has only four items for sale - item A, B, C, and D. The prices of these items are $10, $20, $30, and $40 respectively. A customer has a budget of $60 and wants to buy as many items as possible. However, the customer can only carry a maximum weight of 6 units. The weights of the items are 2, 3, 4, and 5 units re... | tiger_lab_math | [
0.036995235830545425,
0.008911230601370335,
-0.03782082721590996,
-0.022655894979834557,
0.04295067489147186,
-0.026695219799876213,
-0.04216430336236954,
0.0544043630361557,
0.0360855758190155,
-0.012215574271976948,
0.01615767553448677,
0.03955509886145592,
-0.05293597653508186,
-0.03857... | 0.816241 | 0.812458 | 0.003783 |
If $\angle A=20^\circ$ and $\angle AFG=\angle AGF,$ then how many degrees is $\angle B+\angle D?$ [asy]
/* AMC8 2000 #24 Problem */
pair A=(0,80), B=(46,108), C=(100,80), D=(54,18), E=(19,0);
draw(A--C--E--B--D--cycle);
label("$A$", A, W);
label("$B$ ", B, N);
label("$C$", shift(7,0)*C);
label("$D$", D, SE);
label("$E$... | tiger_lab_math | [
0.00855912547558546,
0.005795503966510296,
-0.04414471611380577,
-0.047389231622219086,
0.012292647734284401,
-0.014825515449047089,
-0.027881598100066185,
0.03759864345192909,
0.02997862920165062,
-0.00040480619645677507,
0.008751722052693367,
0.018772976472973824,
-0.049212194979190826,
... | 0.821213 | 0.815555 | 0.005658 |
Consider the following two statements related to abstract algebra: Statement 1: In any finite group, the order of the group is equal to the product of the orders of its elements. Statement 2: If G is a group and H is a normal subgroup of G, then the factor group G/H is abelian if and only if G is abelian. Answer Choi... | tiger_lab_math | [
0.04514123499393463,
0.03249291703104973,
-0.030096787959337234,
-0.04562690109014511,
0.042265310883522034,
-0.04923360049724579,
-0.015891872346401215,
0.06990587711334229,
0.040607891976833344,
-0.02012421190738678,
-0.008267439901828766,
0.019008183851838112,
-0.022290706634521484,
-0.... | 0.810385 | 0.805623 | 0.004762 |
Q is the set of the first n positive odd numbers, where n is a positive integer. Given that n > k, where k is also a positive integer, x is the maximum value of the sum of k distinct members of Q, and y is the minimum value of the sum of k distinct members of Q, what is x + y?
Answer Choices: (A) kn (B) kn + k^2 (C) ... | tiger_lab_math | [
0.033057328313589096,
0.029352689161896706,
-0.04596450552344322,
-0.03939707577228546,
0.04480865225195885,
-0.02611152082681656,
-0.008101018145680428,
0.03174465149641037,
0.05625072494149208,
-0.006451031193137169,
0.026288699358701706,
0.03611999750137329,
-0.05672654137015343,
-0.019... | 0.837721 | 0.835884 | 0.001837 |
Consider the differential equation y' = 2xy with initial condition y(0) = 3. Find the particular solution of this differential equation that satisfies the initial condition. | tiger_lab_math | [
0.006474383641034365,
0.015359721146523952,
-0.03982241451740265,
-0.041854433715343475,
0.03184066340327263,
-0.013781607151031494,
-0.008363543078303337,
0.08597610890865326,
0.04276210069656372,
-0.025794779881834984,
0.03263504430651665,
0.005335221998393536,
-0.036444708704948425,
-0.... | 0.813752 | 0.812353 | 0.001399 |
Let A = {1, 2, 3} and B = {2, 3, 4}. Find the sets (AΔB) ∩ {2, 3, 5} and (A ∩ {2, 3, 5}) Δ (B ∩ {2, 3, 5}). Verify if the symmetric difference of two sets is distributive over intersection. | tiger_lab_math | [
0.04778311029076576,
0.0028478316962718964,
-0.03012891113758087,
-0.015291457064449787,
0.02711329609155655,
-0.010721510276198387,
-0.0074309795163571835,
0.05435555800795555,
0.06792280822992325,
-0.01291457936167717,
0.013504093512892723,
0.015939077362418175,
-0.0309852734208107,
-0.0... | 0.836174 | 0.81623 | 0.019945 |
Let $a,$ $b,$ and $c$ be positive real numbers. Find the minimum value of
\[\frac{a}{b} + \frac{b}{c} + \frac{c}{a}.\] | tiger_lab_math | [
0.02367917262017727,
0.029621366411447525,
-0.04678738862276077,
-0.026368051767349243,
0.04165994003415108,
-0.014275837689638138,
-0.0008617579587735236,
0.05300837382674217,
0.052725061774253845,
-0.03399992734193802,
-0.0026451253797858953,
0.02174319326877594,
-0.022235678508877754,
-... | 0.841297 | 0.838357 | 0.00294 |
A researcher conducted a survey of 100 people to determine the factors that influence their decision to buy a particular brand of mobile phone. From the survey, she identified 7 different factors that were important to consumers. Using factor analysis, she wants to identify the underlying factors that explain most of t... | tiger_lab_math | [
0.02372613735496998,
0.024891003966331482,
-0.03229016810655594,
-0.04629289731383324,
0.039001718163490295,
-0.021063586696982384,
-0.017403120175004005,
0.047505829483270645,
0.03069000132381916,
0.01868269592523575,
0.0004281786677893251,
0.02641603909432888,
-0.04296077787876129,
-0.00... | 0.766289 | 0.762881 | 0.003408 |
Here's a numerical problem related to Nonlinear programming:
A manufacturing company produces two products, Product A and Product B. The production of each product requires machine hours: 3 for Product A and 4 for Product B. The company has a total of 240 machine hours available for production. The profit for each uni... | tiger_lab_math | [
0.004482069052755833,
-0.006741522345691919,
-0.04976107180118561,
-0.014252853579819202,
0.032924287021160126,
0.011851629242300987,
-0.038856662809848785,
0.0727057233452797,
0.030961522832512856,
0.00046823619049973786,
-0.0005758602055720985,
0.02336050756275654,
-0.007505589164793491,
... | 0.813663 | 0.801865 | 0.011798 |
Evaluate the integral of f(z) = z^3 - 2z + 1 over the curve C which is the circle centered at 2+3i with radius 5, using Cauchy's theorem. | tiger_lab_math | [
0.028960254043340683,
0.031336452811956406,
-0.03681434690952301,
-0.029209079220891,
0.03494495898485184,
-0.008905970491468906,
-0.017359578981995583,
0.0439467690885067,
0.02305745519697666,
-0.006799925584346056,
0.029876118525862694,
0.013111114501953125,
-0.034889861941337585,
-0.017... | 0.824591 | 0.822103 | 0.002488 |
Let X be a non-singular projective plane curve of degree 4, defined over the complex numbers. Let D be a divisor on X, with degree 5, and such that the linear system|D| has dimension 1. Find the genus of X. | tiger_lab_math | [
-0.01733388565480709,
0.01716361567378044,
-0.0438833124935627,
-0.06233949586749077,
0.03965736925601959,
-0.0179899949580431,
-0.02199624851346016,
0.041302334517240524,
0.043968021869659424,
-0.021060368046164513,
0.027805957943201065,
0.04223575443029404,
-0.02604842372238636,
-0.01035... | 0.826114 | 0.811087 | 0.015027 |
Let $f(x) = 4x^4+12x^3-9x^2+x+3$ and $d(x) = x^2+3x-2$. If $f(x) = q(x)d(x) + r(x)$ for some polynomials $q(x)$ and $r(x)$ where $\deg r < \deg d$, calculate $q(1)+r(-1)$. Let's write a program. | tiger_lab_math | [
0.03248477354645729,
0.026631366461515427,
-0.05280587822198868,
-0.03862264007329941,
0.03234033286571503,
-0.026526568457484245,
0.001627230434678495,
0.07059958577156067,
0.039878543466329575,
-0.004077092744410038,
0.010423731990158558,
0.03436165675520897,
-0.037558700889348984,
-0.03... | 0.844785 | 0.835157 | 0.009628 |
Given that $x+\sin y=2008$ and $x+2008 \cos y=2007$, where $0 \leq y \leq \frac{\pi}{2}$, find the value of $x+y$. | tiger_lab_math | [
0.01303138118237257,
0.028602633625268936,
-0.05556591600179672,
-0.04003949090838432,
0.029462307691574097,
0.005491114221513271,
-0.0203749630600214,
0.06503508985042572,
0.05067978426814079,
0.0017962446436285973,
0.028605803847312927,
0.047364491969347,
-0.043291740119457245,
-0.025417... | 0.826016 | 0.824595 | 0.001421 |
A manufacturing company needs to decide how many units of a certain product to produce in order to maximize profits. The cost of producing each unit is given by the function C(x) = 100x + 100/x, where x is the number of units produced. The selling price P(x) is given by the function P(x) = 300 - 2x. How many units shou... | tiger_lab_math | [
0.02103324420750141,
0.01290559396147728,
-0.07135336101055145,
0.004414576105773449,
0.025896728038787842,
-0.02522098459303379,
-0.04105904698371887,
0.07346580177545547,
0.040699489414691925,
0.0033402908593416214,
0.003454602789133787,
0.04149705916643143,
-0.00744158960878849,
-0.0272... | 0.811453 | 0.804502 | 0.006951 |
What is the area of the largest circular disk that can be enclosed within the Mandelbrot set? (The Mandelbrot set and all its properties are defined by the mathematical formulas governing the set in Fractal geometry.) | tiger_lab_math | [
0.013033212162554264,
0.04251817241311073,
-0.033395152539014816,
-0.05224784463644028,
0.023599762469530106,
-0.015591372735798359,
-0.028741218149662018,
0.03590951859951019,
0.05446753650903702,
-0.022273316979408264,
0.012744048610329628,
0.027829160913825035,
-0.032035473734140396,
-0... | 0.807949 | 0.798801 | 0.009148 |
The integer $n$ is the largest positive multiple of $15$ such that every digit of $n$ is either $8$ or $0$. Compute $\frac{n}{15}$.
| tiger_lab_math | [
0.02422330714762211,
0.019692901521921158,
-0.028071021661162376,
-0.041702043265104294,
0.046537868678569794,
-0.03089023008942604,
-0.019290804862976074,
0.04789463430643082,
0.03289005532860756,
-0.015625936910510063,
0.027978727594017982,
0.03551506623625755,
-0.045233242213726044,
-0.... | 0.845087 | 0.840444 | 0.004643 |
Suppose that $\alpha$ is inversely proportional to $\beta$. If $\alpha = 4$ when $\beta = 9$, find $\alpha$ when $\beta = -72$. | tiger_lab_math | [
0.018645908683538437,
0.021282415837049484,
-0.04634077474474907,
-0.03897174075245857,
0.025920217856764793,
-0.015338562428951263,
-0.009652872569859028,
0.058579377830028534,
0.05216153711080551,
-0.008959363214671612,
0.018490111455321312,
0.03529804199934006,
-0.06592193990945816,
0.0... | 0.842512 | 0.840605 | 0.001907 |
Suppose you are driving a car and you travel 300 miles in 5 hours. Using the mean value theorem, determine at least one point in time where your speedometer reads exactly 70 mph. | tiger_lab_math | [
0.025713220238685608,
0.0008360936772078276,
-0.031169280409812927,
-0.037035293877124786,
0.015557799488306046,
-0.009869235567748547,
-0.054130103439092636,
0.05707801878452301,
0.045994020998477936,
-0.025360291823744774,
0.013137755915522575,
0.017131760716438293,
-0.025026172399520874,
... | 0.824657 | 0.823654 | 0.001004 |
There are 12 ordered pairs of integers $(x,y)$ that satisfy $x^2 + y^2 = 25$. What is the greatest possible sum $x+y$? Let's write a program. | tiger_lab_math | [
0.02896658331155777,
0.017051836475729942,
-0.048218972980976105,
-0.03796374425292015,
0.03937697410583496,
-0.02544747292995453,
-0.01927250437438488,
0.04489506781101227,
0.04192390292882919,
-0.030326610431075096,
0.02684173732995987,
0.04628623276948929,
-0.0450374074280262,
-0.026963... | 0.837791 | 0.834625 | 0.003167 |
Consider the elliptic curve given by the equation $y^2=x^3+8x+6$ over the field $\mathbb{Q}$. Find the birational equivalence class of this curve and determine whether it is rational. | tiger_lab_math | [
-0.0019553247839212418,
0.032261237502098083,
-0.02630361169576645,
-0.042732805013656616,
0.03152260184288025,
-0.03903811052441597,
0.004277478903532028,
0.0640130341053009,
0.04529879614710808,
-0.04425102472305298,
0.006811428349465132,
0.018167221918702126,
-0.046007174998521805,
-0.0... | 0.837281 | 0.822942 | 0.014339 |
Consider a dataset containing the following 6 observations: (1, 2), (3, 4), (5, 6), (2, 2), (4, 4), and (6, 6). Using the Manhattan distance metric, perform two-cluster hierarchical clustering on the dataset. What are the resulting two clusters? | tiger_lab_math | [
0.012939630076289177,
0.014626316726207733,
-0.03237200528383255,
-0.030412854626774788,
0.04058143496513367,
-0.01312051247805357,
-0.043164126574993134,
0.0434814877808094,
0.051895275712013245,
-0.022488219663500786,
0.015523955225944519,
0.0031754234805703163,
-0.02708309330046177,
-0.... | 0.809248 | 0.808392 | 0.000855 |
Consider the topological space (X,τ) where X={1,2,3,4} and τ is the topology given by τ={∅,{1},{1,2,3},{1,2,3,4}}. Determine whether this topological space satisfies the T1 separation axiom. | tiger_lab_math | [
0.037751179188489914,
0.01871640235185623,
-0.04583663120865822,
-0.02575654163956642,
0.0541507825255394,
-0.026414716616272926,
-0.020398491993546486,
0.05668989196419716,
0.046059925109148026,
-0.009414639323949814,
0.012783943675458431,
0.015651915222406387,
-0.03322350233793259,
-0.01... | 0.824703 | 0.804339 | 0.020364 |
Using Hill cipher encryption with a 2x2 matrix key, encrypt the message "MATH" where A=0, B=1, C=2, D=3. The key is:
|2 3|
|1 4| | tiger_lab_math | [
0.04892617091536522,
0.02615596354007721,
-0.024979781359434128,
-0.04985181987285614,
0.03608579561114311,
-0.03704184293746948,
-0.03532855212688446,
0.04792223870754242,
0.03882346302270889,
-0.037627022713422775,
0.017792271450161934,
0.018199339509010315,
-0.023367466405034065,
-0.018... | 0.819407 | 0.817457 | 0.00195 |
Consider the reaction-diffusion equation:
$\frac{\partial u}{\partial t} = r \cdot \Delta u + k \cdot u - u^3$
where $u$ is the concentration of a chemical substance, $r$ is the diffusion coefficient, $k$ is the reaction rate constant, and $u^3$ represents a nonlinear reaction term.
A math student needs to find the ... | tiger_lab_math | [
0.0006924236076883972,
0.02759118378162384,
-0.04674089327454567,
-0.036673374474048615,
0.047108061611652374,
-0.016362730413675308,
-0.03238463029265404,
0.09655012935400009,
0.039846789091825485,
-0.012735011987388134,
0.011857595294713974,
0.018614178523421288,
-0.023836370557546616,
-... | 0.813939 | 0.79652 | 0.017419 |
Consider the space curve given by the vector equation r(t) = <cos(t), sin(t), t>. Find the curvature and torsion of the curve at the point corresponding to t = π/2. | tiger_lab_math | [
0.013138972222805023,
0.034806255251169205,
-0.04025601968169212,
-0.0335693433880806,
0.033639367669820786,
-0.013121794909238815,
-0.02082604356110096,
0.06407306343317032,
0.026657316833734512,
-0.016439931467175484,
0.01922614499926567,
0.00989455170929432,
-0.025272171944379807,
0.001... | 0.816226 | 0.810212 | 0.006014 |
Given a noncommutative algebra A with generators x and y, and relations x^2 = y^2 = 0, xyx = x, yxy = y, compute the dimension of the noncommutative affine scheme Spec(A). | tiger_lab_math | [
0.040854018181562424,
0.009864913299679756,
-0.03568115830421448,
-0.05322403088212013,
0.019969770684838295,
-0.031757839024066925,
-0.024654144421219826,
0.04448028653860092,
0.032743774354457855,
-0.01773056946694851,
0.019551489502191544,
0.035764288157224655,
-0.04156799241900444,
-0.... | 0.817987 | 0.81567 | 0.002317 |
there are 3 prizes to be distributed among 10 students . if no students gets more than one prize , then this can be done in ? Please write a program to solve it | tiger_lab_math | [
0.00831759162247181,
0.011215724050998688,
-0.024553941562771797,
-0.023185428231954575,
0.026507485657930374,
-0.041211552917957306,
-0.022797388955950737,
0.06749529391527176,
0.008637117221951485,
-0.03153420612215996,
0.03850054740905762,
0.03865385428071022,
-0.03768372908234596,
-0.0... | 0.825614 | 0.81924 | 0.006374 |
Pyramid $OABCD$ has square base $ABCD,$ congruent edges $\overline{OA}, \overline{OB}, \overline{OC},$ and $\overline{OD},$ and $\angle AOB=45^\circ.$ Let $\theta$ be the measure of the dihedral angle formed by faces $OAB$ and $OBC.$ Given that $\cos \theta=m+\sqrt{n},$ where $m$ and $n$ are integers, find $m+n.$
| tiger_lab_math | [
0.022066522389650345,
0.021891672164201736,
-0.048256319016218185,
-0.0500473752617836,
0.026982402428984642,
-0.017378052696585655,
-0.015496409498155117,
0.05806932598352432,
0.020243149250745773,
0.006844413001090288,
0.028389476239681244,
0.017436007037758827,
-0.03630710020661354,
-0.... | 0.839371 | 0.813499 | 0.025873 |
abcd is a square where ab = √ 2016 . let x be a point on ab and y be a point on cd such that ax = cy . compute the area of trapezoid axyd . Please respond by writing a program in Python. | tiger_lab_math | [
0.025242703035473824,
0.012216218747198582,
-0.043198760598897934,
-0.042846158146858215,
0.04322357475757599,
-0.04887918010354042,
-0.009740344248712063,
0.048018451780080795,
0.04607618972659111,
0.0004645339213311672,
0.028096674010157585,
0.03666084632277489,
-0.03641291707754135,
-0.... | 0.814887 | 0.810365 | 0.004522 |
Assuming that the metric is g = dx^2 + dy^2 on the Euclidean plane, find all isometries that preserve the orientation of the plane. | tiger_lab_math | [
0.008661559782922268,
0.021034926176071167,
-0.03686998412013054,
-0.05523855239152908,
0.033626314252614975,
-0.032273635268211365,
-0.020118743181228638,
0.04396285489201546,
0.05070263519883156,
-0.039250656962394714,
0.002732740016654134,
0.025857144966721535,
-0.011639013886451721,
-0... | 0.8315 | 0.822108 | 0.009392 |
a satellite is composed of 30 modular units , each of which is equipped with a set of sensors , some of which have been upgraded . each unit contains the same number of non - upgraded sensors . if the number of non - upgraded sensors on one unit is 1 / 6 the total number of upgraded sensors on the entire satellite , wh... | tiger_lab_math | [
0.0061088427901268005,
0.028335880488157272,
-0.05081576108932495,
-0.02799474261701107,
0.05719982087612152,
-0.03222009912133217,
-0.013000678271055222,
0.07522399723529816,
0.019794583320617676,
-0.018130850046873093,
0.025112811475992203,
0.0279987882822752,
-0.03576648607850075,
-0.03... | 0.822784 | 0.818085 | 0.004699 |
Suppose that Alice and Bob want to do the Diffie-Hellman key exchange using a prime number p = 11 and a primitive root g = 2. Alice chooses a secret number a = 4 and computes g^a mod p. Bob chooses a secret number b = 3 and computes g^b mod p. Compute the secret key that Alice and Bob can agree upon. | tiger_lab_math | [
0.042585406452417374,
0.041403550654649734,
-0.03102642297744751,
-0.05483242869377136,
0.02521463856101036,
-0.049654219299554825,
-0.008100375533103943,
0.035121262073516846,
0.040773309767246246,
-0.007412631064653397,
0.027663355693221092,
0.015598519705235958,
-0.042310405522584915,
-... | 0.831002 | 0.826795 | 0.004207 |
Compute the first cohomology group $H^1(\mathbb{R}^2\setminus\{(0,0)\},\mathbb{Z})$ of the complement of the origin in $\mathbb{R}^2$ with coefficients in integers $\mathbb{Z}$. | tiger_lab_math | [
0.006634915247559547,
0.04302772507071495,
-0.01432853564620018,
-0.03977624326944351,
0.031219201162457466,
-0.019709158688783646,
0.0028151453007012606,
0.04641912877559662,
0.04694512113928795,
-0.02371346950531006,
0.007787402719259262,
0.019393160939216614,
-0.023113874718546867,
-0.0... | 0.825922 | 0.814176 | 0.011747 |
In the diagram, $\triangle ABE$, $\triangle BCE$ and $\triangle CDE$ are right-angled, with $\angle AEB=\angle BEC = \angle CED = 60^\circ$, and $AE=24$. [asy]
pair A, B, C, D, E;
A=(0,20.785);
B=(0,0);
C=(9,-5.196);
D=(13.5,-2.598);
E=(12,0);
draw(A--B--C--D--E--A);
draw(B--E);
draw(C--E);
label("A", A, N);
label("B",... | tiger_lab_math | [
0.0346120223402977,
0.024189379066228867,
-0.03658212348818779,
-0.03690489009022713,
0.02684261091053486,
-0.035931188613176346,
-0.022352449595928192,
0.053684014827013016,
0.0384221114218235,
0.008853893727064133,
0.014458250254392624,
0.027657251805067062,
-0.010910379700362682,
-0.020... | 0.833464 | 0.813953 | 0.019512 |
$A$, $B$, $C$, and $D$ are points on a circle, and segments $\overline{AC}$ and $\overline{BD}$ intersect at $P$, such that $AP=8$, $PC=1$, and $BD=6$. Find $BP$, given that $BP < DP.$
[asy]
unitsize(0.6 inch);
draw(circle((0,0),1));
draw((-0.3,0.94)--(0.3,-0.94));
draw((-0.7,-0.7)--(0.7,-0.7));
label("$A$",(-0.3,0... | tiger_lab_math | [
0.019491927698254585,
0.012904803268611431,
-0.042929161339998245,
-0.022156208753585815,
0.03544324263930321,
-0.02814151719212532,
-0.02419107034802437,
0.04088041931390762,
0.047645751386880875,
0.0053320955485105515,
0.02077144756913185,
0.027007898315787315,
-0.016336262226104736,
-0.... | 0.836019 | 0.818607 | 0.017411 |
Consider the curve C defined by the following parametric equations:
x = cos(t)
y = sin(t)
z = 2t
Find the curvature and torsion of C at t=π/4. | tiger_lab_math | [
0.030292212963104248,
0.04029344394803047,
-0.03573087975382805,
-0.016134066507220268,
0.043988388031721115,
-0.02289576828479767,
-0.018269646912813187,
0.053633179515600204,
0.027816247195005417,
-0.012095687910914421,
0.013662668876349926,
0.020932022482156754,
-0.027534600347280502,
0... | 0.826133 | 0.821323 | 0.00481 |
Consider the noncommutative polynomial ring $k<x,y>$ over the field $k$. Let $A$ be an algebra in this ring defined by the relations $xy = 2yx$ and $y^2 = 0$. Find the centralizer of $A$ in this ring, i.e., the set of all elements $f \in k<x,y>$ such that $fa = af$ for all $a \in A$. | tiger_lab_math | [
0.05332660675048828,
0.03346269205212593,
-0.04963313415646553,
-0.039567142724990845,
0.019315049052238464,
-0.02012351155281067,
0.009899858385324478,
0.03621955215930939,
0.03945141285657883,
-0.013716493733227253,
0.02014988847076893,
0.019826604053378105,
-0.04045649245381355,
0.00115... | 0.832562 | 0.803902 | 0.028661 |
Find all real numbers $k$ such that
\[\left\| k \begin{pmatrix} 2 \\ -3 \end{pmatrix} - \begin{pmatrix} 4 \\ 7 \end{pmatrix} \right\| = 2 \sqrt{13}.\]Enter all the solutions, separated by commas. Please respond by writing a program in Python. | tiger_lab_math | [
0.0431702695786953,
0.04924418777227402,
-0.051314279437065125,
-0.05067955330014229,
0.050826434046030045,
-0.013616357930004597,
-0.008417963981628418,
0.03833550587296486,
0.039116136729717255,
-0.011470302939414978,
0.02085978351533413,
0.04107547923922539,
-0.031280938535928726,
-0.02... | 0.839653 | 0.835607 | 0.004046 |
A manufacturing company is trying to minimize the cost of transporting goods from their warehouse to the retail stores. The distance between the warehouse and the stores are given in the table below:
| Store | Distance from Warehouse (km) |
|-------|----------------------------|
| A | 10 |
... | tiger_lab_math | [
0.017188720405101776,
0.006091686896979809,
-0.03237266093492508,
-0.013012619689106941,
0.021098485216498375,
-0.019175805151462555,
-0.06428851187229156,
0.04208541288971901,
0.03952930495142937,
-0.010709024034440517,
0.02186649665236473,
0.03446919843554497,
-0.030457517132163048,
-0.0... | 0.797412 | 0.794478 | 0.002934 |
Let A={1,2,3,4} and B={3,4,5,6}. Find (A Δ B) ∪ A, where Δ represents the symmetric difference between set A and set B. | tiger_lab_math | [
0.035909686237573624,
0.00885365903377533,
-0.03384091332554817,
-0.018154066056013107,
0.04134822264313698,
-0.021828576922416687,
-0.022281955927610397,
0.05454516038298607,
0.06628214567899704,
-0.024452365934848785,
0.01595909334719181,
0.03156433627009392,
-0.03114647977054119,
-0.010... | 0.842762 | 0.838354 | 0.004408 |
Consider the following network with six nodes and eight directed edges, given in the table below:
| Edge | Capacity |
|------|----------|
| s -> 1 | 3 |
| s -> 2 | 2 |
| 1 -> 2 | 1 |
| 1 -> 3 | 3 |
| 2 -> 4 | 1 |
| 3 -> 2 | 1 |
| 3 -> 5 | 2 |
| 4 -> t | 2 |
| 5 -> t | 3 |
Find the maximum flow that can be sent from t... | tiger_lab_math | [
0.03630652651190758,
0.022841380909085274,
-0.02522185817360878,
-0.00806995015591383,
0.02603842131793499,
-0.03377222642302513,
-0.05735514312982559,
0.06365608423948288,
0.031675856560468674,
-0.004199448972940445,
0.029925527051091194,
0.03913212940096855,
-0.02935810573399067,
-0.0186... | 0.810411 | 0.802177 | 0.008234 |
Here is a numerical problem for a math student to solve using firefly algorithm:
A coffee shop owner wants to maximize their profits by determining the optimal prices of their specialty drinks, Cappuccino and Latte. The cost of producing a Cappuccino and Latte is $1.50 and $1.75 respectively. The market research shows... | tiger_lab_math | [
0.0173040758818388,
0.012273614294826984,
-0.03372439742088318,
-0.04029316082596779,
0.043708931654691696,
-0.01450799684971571,
-0.032175250351428986,
0.06721680611371994,
0.044731561094522476,
-0.011712293140590191,
0.006797335110604763,
0.008490397594869137,
-0.03920809552073479,
-0.02... | 0.81943 | 0.803119 | 0.016311 |
Statement 1| The intersection of two subgroups of a group is again a subgroup. Statement 2 | A finite group with an even number of elements must have a subgroup of order 2. Answer Choices: (A) True, True (B) False, False (C) True, False (D) False, True | tiger_lab_math | [
0.05172129347920418,
0.01641155034303665,
-0.02896784618496895,
-0.04026266187429428,
0.03543936461210251,
-0.0287461020052433,
-0.0189136303961277,
0.04949453845620155,
0.03634706884622574,
-0.01036599837243557,
0.02028830163180828,
0.012366805225610733,
-0.019037459045648575,
-0.01887653... | 0.812795 | 0.805916 | 0.006879 |
Here's a numerical problem related to RSA encryption:
Bob wants to send a confidential message to Alice using RSA encryption. Bob chooses p = 17 and q = 23 as his primes, and his public key is (n, e) = (391, 7). To encrypt the message, he breaks it up into blocks of two digits each, and converts each block to a number... | tiger_lab_math | [
0.03889528661966324,
0.0403539314866066,
-0.03410118445754051,
-0.046263135969638824,
0.03167654573917389,
-0.014996492303907871,
0.00013862618652638048,
0.040956079959869385,
0.018618669360876083,
-0.011201314628124237,
0.028018826618790627,
0.02401633933186531,
-0.04657723382115364,
-0.0... | 0.827783 | 0.822401 | 0.005382 |
A company manufactures two types of products, A and B. The profit per unit for product A is $10 and for product B is $15. The company has 60 units of raw material to produce these products. Product A requires 3 units of raw material per unit, while product B requires 5 units of raw material per unit. The company has a ... | tiger_lab_math | [
0.03696785494685173,
0.01353997178375721,
-0.04689522087574005,
-0.006834679748862982,
0.016139954328536987,
-0.034806083887815475,
-0.0423586443066597,
0.09142166376113892,
0.021677836775779724,
-0.01900632679462433,
0.008169947192072868,
0.03633774816989899,
-0.012423308566212654,
-0.036... | 0.799693 | 0.796748 | 0.002946 |
Let $p(x) = x^{2008} + x^{2007} + x^{2006} + \cdots + x + 1,$
and let $r(x)$ be the polynomial remainder when $p(x)$ is divided by $x^4+x^3+2x^2+x+1$. Find the remainder when $|r(2008)|$ is divided by $1000$.
| tiger_lab_math | [
0.019998855888843536,
0.029534071683883667,
-0.03784448280930519,
-0.025984158739447594,
0.015691211447119713,
-0.03786091506481171,
-0.025397926568984985,
0.07849635928869247,
0.04983796551823616,
-0.005196418147534132,
0.007818346843123436,
0.03170459717512131,
-0.04033353552222252,
-0.0... | 0.84629 | 0.837273 | 0.009018 |
Statement 1 | The set of all integers under addition is an example of an abelian group. Statement 2 | Every finite integral domain is a field. Answer Choices: (A) True, True (B) False, False (C) True, False (D) False, True, | tiger_lab_math | [
0.03107898123562336,
0.02539793588221073,
-0.02117675542831421,
-0.04792090132832527,
0.04688778147101402,
-0.049222320318222046,
-0.0004393654817249626,
0.0396890714764595,
0.04796770215034485,
-0.019910523667931557,
0.004518960136920214,
0.025497158989310265,
-0.025402123108506203,
-0.01... | 0.815613 | 0.811475 | 0.004138 |
Here's a numerical optimization problem using particle swarm optimization:
Find the minimum value of the function f(x) = x^4 - 3x^3 + 2, given that x belongs to the interval [-5, 5]. Use particle swarm optimization with a swarm size of 20 and maximum number of iterations of 100. Additionally, set the inertia weight t... | tiger_lab_math | [
0.008654952980577946,
-0.0017688064835965633,
-0.043460361659526825,
-0.0361710749566555,
0.0347420871257782,
-0.029508473351597786,
-0.03654685616493225,
0.05561637133359909,
0.053239837288856506,
0.00522023718804121,
0.008695199154317379,
0.014616879634559155,
-0.019208593294024467,
-0.0... | 0.81723 | 0.816325 | 0.000904 |
Consider a complex projective curve $C$ defined by the equation $y^2 = x^3 + ax+b$, where $a,b \in \mathbb{C}$ and $4a^3 + 27b^2 \neq 0$. Let $\mathcal{O}_C(n)$ be the line bundle on $C$ associated to the divisor $n \cdot \operatorname{point}(P)$, where $P$ is a fixed point on $C$ and $n \in \mathbb{Z}$. Compute $H^1(C... | tiger_lab_math | [
0.016599025577306747,
0.04186059907078743,
-0.02751963399350643,
-0.050916947424411774,
0.0364983044564724,
-0.02632642723619938,
-0.008237481117248535,
0.060109056532382965,
0.04937481880187988,
-0.010324114002287388,
0.0003782685671467334,
0.03750235214829445,
-0.025987109169363976,
-0.0... | 0.820832 | 0.793668 | 0.027164 |
Let M be a 3-dimensional manifold with metric tensor g given by
\begin{align*}
g_{11}&=4, \hspace{5mm} g_{12}=0, \hspace{5mm} g_{13}=0, \\
g_{22}&=4 \sin^2{u}, \hspace{5mm} g_{23}=0, \\
g_{33}&=1,
\end{align*}
where u is one of the coordinates on the manifold. Find the components of the Riemann curvature tensor of M. ... | tiger_lab_math | [
0.012315399944782257,
0.039997026324272156,
-0.018777094781398773,
-0.07122388482093811,
0.03888401761651039,
-0.011477160267531872,
-0.01825547032058239,
0.06659246236085892,
0.015212183818221092,
0.004750094376504421,
-0.0013601197861135006,
0.030791936442255974,
-0.05536997318267822,
-0... | 0.813976 | 0.800176 | 0.0138 |
What is the conformal mapping from the upper half plane $\mathbb{H}$ to the unit disk $\mathbb{D}$, given by $f(z) = \frac{z-i}{z+i}$? Use the mapping to find the image of the line segment $[-1,1]$ in the upper half plane. | tiger_lab_math | [
0.01854890026152134,
0.016182037070393562,
-0.04402672126889229,
-0.039350081235170364,
0.03297264873981476,
-0.0077321589924395084,
0.010412575677037239,
0.03897677734494209,
0.04339975863695145,
-0.002006220631301403,
0.026183106005191803,
0.020450236275792122,
-0.008320587687194347,
-0.... | 0.81712 | 0.812393 | 0.004727 |
Find $XY$ in the triangle below.
[asy]
unitsize(1inch);
pair P,Q,R;
P = (0,0);
Q= (1,0);
R = (0,1);
draw (P--Q--R--P,linewidth(0.9));
draw(rightanglemark(Q,P,R,3));
label("$X$",P,S);
label("$Y$",Q,S);
label("$Z$",R,N);
label("$12\sqrt{2}$",(Q+R)/2,NE);
label("$45^\circ$",(0.7,0),N);
[/asy] Please respond by writing a... | tiger_lab_math | [
0.02841227874159813,
0.0396517775952816,
-0.04401221126317978,
-0.04899346083402634,
0.03651375323534012,
-0.029533017426729202,
-0.017494164407253265,
0.04097171500325203,
0.04342855513095856,
-0.008327365852892399,
0.04153267294168472,
0.03754603862762451,
-0.006938247941434383,
-0.01963... | 0.836372 | 0.829548 | 0.006824 |
The parabola with equation $y=ax^2+bx+c$ is graphed below:
[asy]
xaxis(-3,7);
yaxis(-5,32);
real g(real x)
{
return 4(x-2)^2-4;
}
draw(graph(g,-1,5));
dot((2,-4));
label("Vertex: $(2,-4)$", (2,-4), SE);
dot((4,12));
label("$(4,12)$", (4,12), E);
[/asy]
The zeros of the quadratic $ax^2 + bx + c$ are at $x=m$ and ... | tiger_lab_math | [
0.02772996947169304,
0.027982698753476143,
-0.04536217823624611,
-0.05082332342863083,
0.0385563001036644,
-0.03067290224134922,
-0.021113285794854164,
0.08700484782457352,
0.040908683091402054,
-0.012051288969814777,
0.0022482560016214848,
0.044443536549806595,
-0.024331919848918915,
-0.0... | 0.833987 | 0.824383 | 0.009604 |
A student wants to buy a notebook that is neither too big nor too small. Using fuzzy logic, create a problem that determines the optimal size of the notebook for the student.
Suppose the student values small notebooks 3 times as much as large notebooks. The maximum notebook size they will accept is 15 inches, and the ... | tiger_lab_math | [
0.0031663852278143167,
-0.00715244747698307,
-0.053803589195013046,
-0.03471098467707634,
0.039222121238708496,
-0.029306894168257713,
-0.020053837448358536,
0.044804587960243225,
0.024219445884227753,
-0.01498662494122982,
0.03906746581196785,
0.042761366814374924,
-0.030573897063732147,
... | 0.817428 | 0.803253 | 0.014175 |
Simplify the following trigonometric expression:
$\frac{\sin^2(x)+\cos^2(x)}{\cos^2(x)-\sin^2(x)}$
where $x$ is an angle in radians. | tiger_lab_math | [
0.0037025457713752985,
0.03756624087691307,
-0.04123743996024132,
-0.04364696145057678,
0.04070194810628891,
-0.02907506562769413,
-0.015498008579015732,
0.05990024656057358,
0.05913756042718887,
-0.0039626117795705795,
0.02159728854894638,
-0.005285749211907387,
-0.018870342522859573,
-0.... | 0.816487 | 0.813514 | 0.002973 |
Consider the metric defined by
$g = 2dx^2 + 2dy^2 + 3dz^2$
a) Find the Killing vector fields that generate the isometries of this metric.
b) Calculate the Lie derivative of the vector field $X = x \frac{\partial}{\partial x} + y \frac{\partial}{\partial y} + z \frac{\partial}{\partial z}$ with respect to the metr... | tiger_lab_math | [
0.02483545057475567,
0.041663385927677155,
-0.04431438818573952,
-0.05612363666296005,
0.03319020941853523,
-0.019151071086525917,
-0.03474126011133194,
0.05323966592550278,
0.031944893300533295,
-0.0049651917070150375,
0.0139211630448699,
0.024667512625455856,
-0.041893623769283295,
-0.01... | 0.821739 | 0.809553 | 0.012186 |
Let $S$ be a smooth, closed surface in $\mathbb{R}^3$ with a Hamiltonian flow $X$. Suppose that the De Rham cohomology group $H^1_{DR}(S)$ is isomorphic to $\mathbb{R}^3$ and that the flow $X$ has three periodic orbits with periods $1$, $2$, and $3$.
What is the sum of the indices of the three periodic orbits of $X$ ... | tiger_lab_math | [
0.008941832929849625,
0.020116794854402542,
-0.05927860736846924,
-0.04275449737906456,
0.02304133214056492,
-0.02075600065290928,
-0.03171220049262047,
0.06488648056983948,
0.02029578946530819,
0.004258315544575453,
0.014142932370305061,
0.026831652969121933,
-0.04067748412489891,
-0.0282... | 0.826737 | 0.816034 | 0.010703 |
Let $a\star b = a^b+ab$. If $a$ and $b$ are positive integers greater than or equal to 2 and $a\star b =15$, find $a+b$. | tiger_lab_math | [
0.039029255509376526,
0.03635334223508835,
-0.02041802741587162,
-0.044193610548973083,
0.04508667811751366,
-0.0011889943853020668,
-0.0047623771242797375,
0.08702358603477478,
0.04280515015125275,
-0.01726546324789524,
0.027201225981116295,
0.033558353781700134,
-0.05551786348223686,
-0.... | 0.844392 | 0.83578 | 0.008612 |
Consider a 2-dimensional surface with a metric given by:
$$ds^2 = (1+3\cos^2 u)d\theta^2+\sin^2 u du^2$$
Find the scalar curvature of this surface at the point $(\theta_0, u_0) = (\pi/2, 0)$. | tiger_lab_math | [
-0.011340154334902763,
0.03413750231266022,
-0.05092846602201462,
-0.05249574035406113,
0.03874482586979866,
-0.01149399671703577,
-0.02031986601650715,
0.06626015156507492,
0.030680973082780838,
-0.00993631687015295,
0.01515746209770441,
0.02440345101058483,
-0.025113562121987343,
-0.0046... | 0.835183 | 0.819047 | 0.016137 |
Suppose there are 10 farmers who share a common pasture. Each farmer can graze up to 10 cows on the pasture. The total carrying capacity of the pasture is 100 cows. If each farmer acts in their own self-interest and grazes as many cows as they can, what is the total number of cows that will be grazed on the pasture? Th... | tiger_lab_math | [
0.021130118519067764,
0.021242709830403328,
-0.049328312277793884,
-0.017458416521549225,
0.011188032105565071,
-0.043993327766656876,
-0.023300983011722565,
0.04854738339781761,
0.023809703066945076,
-0.024672355502843857,
-0.02755035273730755,
0.022733407095074654,
-0.03801344707608223,
... | 0.807529 | 0.794361 | 0.013168 |
Consider a two-dimensional rectangular box with dimensions $4 \times 2$. The voltage at the top side is fixed at $V=5$ and the voltage at the bottom side is fixed at $V=0$. The left and right sides of the box are electrically insulated. Find the potential function $U(x,y)$ inside the box satisfying Laplace's equation f... | tiger_lab_math | [
0.010899936780333519,
0.05140295252203941,
-0.03608480095863342,
-0.04915342479944229,
0.040050141513347626,
-0.026026388630270958,
-0.025941181927919388,
0.06926436722278595,
0.03896767646074295,
0.008083829656243324,
0.024820823222398758,
0.03821946680545807,
-0.019098825752735138,
-0.01... | 0.825906 | 0.805947 | 0.01996 |
Here is a numerical problem for the student to solve:
A company produces two types of products A and B. The profit earned from selling one unit of A is $10, and the profit earned from selling one unit of B is $15. The company has two machines M1 and M2, and each product requires a certain amount of time to be produced... | tiger_lab_math | [
0.018043000251054764,
0.004862441215664148,
-0.043031081557273865,
-0.004523180890828371,
0.01652471534907818,
-0.007913115434348583,
-0.055989690124988556,
0.07569357752799988,
0.04215513542294502,
-0.0026833810843527317,
0.0005445764400064945,
0.03701350837945938,
-0.012649789452552795,
... | 0.822268 | 0.805792 | 0.016476 |
Let $M$ be a smooth manifold of dimension $n$ with a Riemannian metric $g$. Let $\alpha$ be a $k$-form on $M$, where $1 \leq k \leq n$. Calculate the Hodge star operator of $\alpha$ with respect to $g$.
As a specific example, let $M$ be the two-dimensional sphere $S^2$ with the standard round metric $g$ of radius $r$.... | tiger_lab_math | [
0.010671591386198997,
0.046532630920410156,
-0.04507821053266525,
-0.05352967977523804,
0.03180260956287384,
-0.014640387147665024,
-0.03405086323618889,
0.0710950568318367,
0.04094656929373741,
-0.012766408734023571,
0.0035212708171457052,
0.04659831151366234,
-0.0315646268427372,
-0.0545... | 0.808528 | 0.791396 | 0.017132 |
Let a be a positive integer. If q is divisible by 2^a and q is also divisible by 3^(2a), then it is possible that q is NOT divisible by
Answer Choices: (A) 6 (B) 3 × 2^a (C) 2 × 3^ (2a) (D) 6^a (E) 6^ (2a) | tiger_lab_math | [
0.04813430830836296,
0.029105382040143013,
-0.00717022642493248,
-0.021999716758728027,
0.045427966862916946,
-0.03414102643728256,
-0.017772382125258446,
0.07232959568500519,
0.030564917251467705,
-0.013765327632427216,
0.023071251809597015,
0.024158988147974014,
-0.046703189611434937,
-0... | 0.824468 | 0.819816 | 0.004652 |
Let $O$ be the origin, and let $(a,b,c)$ be a fixed point. A plane passes through $(a,b,c)$ and intersects the $x$-axis, $y$-axis, and $z$-axis at $A,$ $B,$ and $C,$ respectively, all distinct from $O.$ Let $(p,q,r)$ be the center of the sphere passing through $A,$ $B,$ $C,$ and $O.$ Find
\[\frac{a}{p} + \frac{b}{q}... | tiger_lab_math | [
0.030072104185819626,
0.04366109147667885,
-0.04992163926362991,
-0.03527645394206047,
0.023365452885627747,
-0.014827102422714233,
-0.015260877087712288,
0.04239566996693611,
0.0473586767911911,
-0.016996122896671295,
0.012153705582022667,
0.030993176624178886,
-0.03316710144281387,
-0.02... | 0.839241 | 0.814959 | 0.024282 |
Consider the moduli space of genus 2 algebraic curves. Let M(2) denote this moduli space.
Find the dimension of M(2) over the field of complex numbers. | tiger_lab_math | [
0.020732831209897995,
0.022447898983955383,
-0.03331272304058075,
-0.062311802059412,
0.03153537958860397,
-0.019983181729912758,
-0.021793894469738007,
0.03691120818257332,
0.03347960487008095,
-0.025785837322473526,
-0.00043215625919401646,
0.035009898245334625,
-0.019420143216848373,
-0... | 0.806733 | 0.79734 | 0.009394 |
Consider the polynomials \[f(x)=1-12x+3x^2-4x^3+5x^4\] and \[g(x)=3-2x-6x^3+9x^4.\] Find $c$ such that the polynomial $f(x)+cg(x)$ has degree 3. | tiger_lab_math | [
0.03369631990790367,
0.02296372689306736,
-0.051326822489500046,
-0.04085490107536316,
0.035512734204530716,
-0.04638676345348358,
-0.013492146506905556,
0.0659543052315712,
0.016044801101088524,
0.0008781612850725651,
0.016837691888213158,
0.03901247680187225,
-0.03813084959983826,
-0.021... | 0.843624 | 0.828537 | 0.015087 |
Consider the following undirected graph with 8 vertices:
```
1
/ \
2 - 3
/ \ / \
4 - 5 - 6
\ /
7
|
8
```
What is the maximum number of independent vertices that can be selected from this graph? | tiger_lab_math | [
0.03482644632458687,
0.027969244867563248,
-0.033739227801561356,
-0.011890347115695477,
0.027826324105262756,
-0.02881099097430706,
-0.03471219167113304,
0.040252722799777985,
0.024761250242590904,
-0.01640474796295166,
0.012041408568620682,
0.030260125175118446,
-0.02048608288168907,
-0.... | 0.829126 | 0.82627 | 0.002857 |
Find the diagonal matrix D, and invertible matrix P, such that P^-1AP = D for the matrix A = [[3, 1], [0, 2]] in linear algebra. | tiger_lab_math | [
0.03006136044859886,
0.02152981609106064,
-0.04391612485051155,
-0.05007125809788704,
0.02938421070575714,
-0.0188271664083004,
-0.007633463945239782,
0.061284229159355164,
0.028970835730433464,
-0.015982309356331825,
0.018665071576833725,
0.015657110139727592,
-0.030318018049001694,
-0.02... | 0.832805 | 0.826213 | 0.006592 |
Consider the surface given by the equation: $z = x^2 - 2y^2$. Calculate the sectional curvature of this surface at the point $(1,-1,0)$ in the direction of the vectors $\vec{v} = \frac{1}{\sqrt{2}}\langle 1, -1,0\rangle$ and $\vec{w} = \frac{1}{\sqrt{2}}\langle 1,1,0\rangle$. | tiger_lab_math | [
0.015773892402648926,
0.04715113714337349,
-0.03984491899609566,
-0.044197749346494675,
0.04648294672369957,
0.003992298152297735,
-0.01126350462436676,
0.06770341098308563,
0.008581820875406265,
-0.015401809476315975,
0.017520055174827576,
0.033971790224313736,
-0.030551783740520477,
-0.0... | 0.824872 | 0.807053 | 0.017819 |
Statement 1| The order of an element in a group is always equal to the order of the group. Statement 2 | In a finite group, the order of an element divides the order of the group. Answer Choices: (A) True, True (B) False, False (C) True, False (D) False, True | tiger_lab_math | [
0.04630891606211662,
0.02840658277273178,
-0.03140981122851372,
-0.05636409670114517,
0.03494585305452347,
-0.051798537373542786,
-0.017305441200733185,
0.0531499907374382,
0.02768779918551445,
-0.02132875844836235,
0.019131703302264214,
0.010014262050390244,
-0.03792691230773926,
0.001539... | 0.8055 | 0.801528 | 0.003972 |
Consider a group of order 8. Let's denote the group by G. Determine which one of the following groups is isomorphic to G: the cyclic group C8, the quaternion group Q8, or the dihedral group D4. | tiger_lab_math | [
0.018525822088122368,
0.016583610326051712,
-0.022198906168341637,
-0.04540296271443367,
0.029508255422115326,
-0.04817935451865196,
-0.014799583703279495,
0.052641239017248154,
0.03874798119068146,
-0.008703529834747314,
0.01847492903470993,
0.007070521358400583,
-0.0448610894382,
-0.0130... | 0.812399 | 0.805385 | 0.007015 |
A quadrilateral has vertices at $(0,1)$, $(3,4)$, $(4,3)$ and $(3,0)$. Its perimeter can be expressed in the form $a\sqrt2+b\sqrt{10}$ with $a$ and $b$ integers. What is the sum of $a$ and $b$? | tiger_lab_math | [
0.015886086970567703,
0.03539799526333809,
-0.03830946609377861,
-0.02594611980021,
0.031183501705527306,
-0.01892852783203125,
-0.03063853457570076,
0.05127041041851044,
0.03913373127579689,
-0.0010553167667239904,
0.014351942576467991,
0.0528724305331707,
-0.03722209855914116,
-0.0330346... | 0.844947 | 0.83443 | 0.010517 |
Steven is preparing a shipment of boxes to deliver to a customer for his chemical supply business. The products are very delicate and must be carefully packed, so partially filled boxes can’t be shipped. Steven has three trucks that can be sent on each delivery. Each truck can carry a load of no more than 2,000 pounds ... | tiger_lab_math | [
0.027443012222647667,
0.032126739621162415,
-0.031086402013897896,
-0.028541425243020058,
0.03564520925283432,
-0.017237866297364235,
-0.032220158725976944,
0.06659569591283798,
0.021683484315872192,
-0.021308399736881256,
0.012863743118941784,
0.04620347544550896,
-0.04344520345330238,
-0... | 0.800086 | 0.793783 | 0.006303 |
Consider an auction where three bidders are participating.
The valuations of the bidders for the item being auctioned are as follows:
Bidder 1: $100
Bidder 2: $80
Bidder 3: $120
The auction follows a second-price sealed-bid rule, which means that the highest bidder wins the item, but pays the price of the second-hig... | tiger_lab_math | [
0.012968383729457855,
0.011190014891326427,
-0.03597014397382736,
-0.01376388780772686,
0.04046596959233284,
-0.02298492193222046,
-0.044719647616147995,
0.048097990453243256,
0.04488246142864227,
-0.013182863593101501,
0.03055376000702381,
0.021859386935830116,
-0.048035286366939545,
-0.0... | 0.800731 | 0.79057 | 0.010161 |
In triangle $ABC,$ $\angle B = 60^\circ$ and $\angle C = 45^\circ.$ The point $D$ divides $\overline{BC}$ in the ratio $1:3$. Find
\[\frac{\sin \angle BAD}{\sin \angle CAD}.\] | tiger_lab_math | [
0.021584276109933853,
0.02168240211904049,
-0.04288274049758911,
-0.03889599069952965,
0.030079921707510948,
-0.0029220145661383867,
-0.001987546682357788,
0.054026249796152115,
0.03299123793840408,
-0.004743472207337618,
0.03951847180724144,
0.019631804898381233,
-0.03468646481633186,
-0.... | 0.835174 | 0.818267 | 0.016907 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.