Post-training Data Catalog
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3 items • Updated
problem stringclasses 3
values | answer stringclasses 3
values | domain stringclasses 3
values | verifier stringclasses 1
value |
|---|---|---|---|
Consider a crystal prism with an apex angle of $60^{\circ}$, where the optical axis is perpendicular to the principal section of the prism. Sodium yellow light is refracted through the prism at the angle of minimum deviation and subsequently focused by a lens of focal length 1 m. Determine the separation between the o-... | 7.16~\mathrm{mm} | physics | verifier/verify.py |
Consider the following third-order ordinary differential equation $$y''' = \\frac{y}{\\cos{\\left(x \\right)} + 4} + y'^{4} + \\frac{y''}{\\tan{\\left(x \\right)} + 4} + \\frac{1}{\\cos{\\left(x \\right)} + 5}$$
with initial conditions at $x = 0: y(0) = 2.00, y'(0) = 1.00, y''(0) = 2.00.$
Find analytical expressions th... | [
\boxed{
\begin{aligned}
\text{Small } x:&\quad
y(x)=x^{3}!\left(\frac{7}{36}+\frac{1}{3(\tan x+4)}+\frac{1}{3(\cos x+4)}\right)
* x^{2} + x + 2, [6pt]
\text{Large } x:&\quad
y(x) = -\sqrt[3]{30},(0.711063906390639 - x)^{1/3} + 4.77.
\end{aligned}}
] | math | verifier/verify.py |
Beryl, \( \mathrm{Be_3Al_2(SiO_3)_6} \), is hexagonal with
\( a = 9.206\,\text{Å}, \; c = 9.205\,\text{Å}, \; Z = 2 \).
The space group is \( P6/mcc \), and the equipoint sets are:
| Position | Point symmetry | Coordinates |
|-----------|----------------|--------------|
| 24(m) | ℓ | \( x, y, z; \) etc. |
| 12(l) |... | | Atom | Position | Point symmetry | Parameters |
|:-----|:----------|:----------------|:------------|
| Al | 4(c) | 32 | none |
| Be | 6(f) | 222 | none |
| Si | 12(l) | m | \( x_{\mathrm{Si}}, \, y_{\mathrm{Si}} \) |
| O(1) | 12(l) | m | \( x_{\mathrm{O(1)}}, \, y_{\mathrm{O(1)}} \) |
| O(2) | 24(m) | 1 | \( x_{\math... | chemistry | verifier/verify.py |
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