instruction_seed stringlengths 56 2.52k | source stringclasses 1
value | embeddings sequencelengths 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 |
|---|---|---|---|---|---|
Find the solution to the boundary value problem:
$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ in the region $0 < x < \pi$, $0 < y < \pi/4$,
subject to the boundary conditions:
$u(0,y) = u(\pi,y) = \sin(y)$,
$u(x,0) = 0$,
$u(x,\pi/4) = \sin(3x)$ | tiger_lab_math | [
-0.010558968409895897,
0.03424713388085365,
-0.04453207552433014,
-0.029888439923524857,
0.052983608096838,
0.01061920914798975,
-0.003305132733657956,
0.06413992494344711,
0.0668499618768692,
0.0012624072842299938,
0.01343163475394249,
0.0411829873919487,
-0.005262796767055988,
-0.0249137... | 0.825875 | 0.802484 | 0.023391 |
A mechanical engineer has to design a cantilever beam with a length of 30 cm and a height of 7 cm using a material with a Young's modulus of 2 x 10^5 MPa. The beam must withstand a uniformly distributed load of 1000 N at the free end.
The cross-section of the beam is a rectangle with a variable width (w) and a consta... | tiger_lab_math | [
0.012581391260027885,
0.028438083827495575,
-0.04592107981443405,
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0.04418232664465904,
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0.08437298238277435,
0.028339484706521034,
-0.0029547226149588823,
0.02180730737745762,
0.03768733888864517,
-0.031005024909973145,
-0.00... | 0.816799 | 0.809957 | 0.006841 |
Find all the fixed points of the action of the group $\mathbb{Z}/3\mathbb{Z}$ on the projective plane $\mathbb{P}^2$ induced by the action given by $\zeta \cdot [x:y:z] = [\zeta x : \zeta^2 y : z]$ for $\zeta \in \mathbb{Z}/3\mathbb{Z}$. | tiger_lab_math | [
0.036154888570308685,
0.03863834589719772,
-0.042351655662059784,
-0.04630023613572121,
0.02749769203364849,
-0.021206289529800415,
0.009128126315772533,
0.03507247194647789,
0.04224829748272896,
-0.011787119321525097,
0.009252382442355156,
0.02693241275846958,
-0.029183536767959595,
-0.02... | 0.839268 | 0.817662 | 0.021606 |
Let $a, b, c$ be complex numbers, with $a$ real, such that \[a+b+c=ab+bc+ca=abc=3.\]Find $a$. | tiger_lab_math | [
0.0430360808968544,
0.03533778712153435,
-0.04192732647061348,
-0.04197528958320618,
0.0310788806527853,
-0.010916762985289097,
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0.05427214130759239,
0.031067976728081703,
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0.020630789920687675,
0.047723837196826935,
-0.04940740019083023,
-0.04692... | 0.849012 | 0.837555 | 0.011457 |
Let A = {1, 2, 3, 4} and B = {2, 3, 4, 5}. Find (A △ B) ∪ B, where △ represents the symmetric difference of two sets. | tiger_lab_math | [
0.03988746181130409,
0.011722471565008163,
-0.03544275462627411,
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0.03511078283190727,
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0.06778208911418915,
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0.009114665910601616,
0.02805485390126705,
-0.02553899772465229,
-0.0050... | 0.833718 | 0.828293 | 0.005424 |
Here's a numerical problem for a math student to solve related to the Chinese Remainder Theorem:
Suppose you are given the system of congruences:
x ≡ 3 (mod 5)
x ≡ 4 (mod 7)
x ≡ 2 (mod 11)
Find the smallest positive integer solution for x using the Chinese Remainder Theorem. | tiger_lab_math | [
0.04030098021030426,
-0.006493410095572472,
-0.03921724110841751,
-0.044335316866636276,
0.04608947038650513,
-0.019395537674427032,
-0.019245987758040428,
0.051809411495923996,
0.043729573488235474,
-0.003569741966202855,
0.00977916270494461,
0.01643378660082817,
-0.03311329707503319,
-0.... | 0.833639 | 0.831702 | 0.001937 |
The length of the longer side of rectangle $R$ is $10$ percent more than the length of a side of square $S.$ The length of the shorter side of rectangle $R$ is $10$ percent less than the length of a side of square $S.$ What is the ratio of the area of rectangle $R$ to the area of square $S?$ Express your answer as a co... | tiger_lab_math | [
0.013441693969070911,
0.018856694921851158,
-0.05987972766160965,
-0.041954003274440765,
0.016056066378951073,
-0.030684826895594597,
-0.02900644950568676,
0.07354069501161575,
0.0463496595621109,
-0.000204465861315839,
0.02269255742430687,
0.030529391020536423,
-0.05489160493016243,
-0.00... | 0.819204 | 0.814341 | 0.004864 |
Suppose a farmer has 400 acres of land to plant two crops: wheat and corn. The profit per acre for growing wheat is $200 and for growing corn is $300. The farmer can spend at most $50,000 on farming operations. Each acre of wheat requires $150 of farming costs and each acre of corn requires $200 of farming costs. Assum... | tiger_lab_math | [
0.01739128679037094,
0.02355325222015381,
-0.03607936576008797,
-0.021936874836683273,
0.004688873887062073,
-0.02541307732462883,
-0.027701035141944885,
0.06220573931932449,
0.02726076729595661,
-0.007285831030458212,
-0.012249589897692204,
0.03553755581378937,
-0.03215960040688515,
-0.02... | 0.792364 | 0.789608 | 0.002756 |
"A math student is asked to determine the symmetries of a 4-dimensional manifold with a metric tenso(...TRUNCATED) | tiger_lab_math | [0.03651687875390053,0.018007639795541763,-0.04093579947948456,-0.0676165521144867,0.042624332010746(...TRUNCATED) | 0.823243 | 0.813679 | 0.009564 |
"If u is the real part of a function, and v is the imaginary part, then the Cauchy-Riemann equations(...TRUNCATED) | tiger_lab_math | [0.036475975066423416,0.03955099359154701,-0.03588329628109932,-0.045038193464279175,0.0239824093878(...TRUNCATED) | 0.814541 | 0.810273 | 0.004269 |
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