Claude: on the 100-problem benchmark

Evaluate \(\underset{x \rightarrow 0}{\mathrm{lim}}\frac{\sin \! \left(x \right)-x +\frac{x^{3}}{6}}{x^{5}}\).

Analysis: Run 1's answer \(-1/120\) is the negative of run 6's answer \(1/120\), so they differ in sign.

For \(f \! \left(x \right) = {\mathrm e}^{x}\) and Taylor expansion centered at 0, find the smallest \(n\) such that the Lagrange remainder satisfies \({| R_{n} \! \left(x \right)|}< 1.0\times 10^{-6}\) for all \(x\) in \([0, 0.5]\).

Analysis: The answers differ in the integer value of n (run 2 gives \(9\), run 1 gives \(7\)), representing different choices for the smallest Taylor expansion order satisfying the remainder bound.

Find the envelope of the one-parameter family of lines \(y = m x +\frac{1}{m}\) for \(0<m\), and identify the resulting curve.

Analysis: The answer by run 1 is the full parabola \(y^2 = 4x\) (both branches \(y = \pm 2\sqrt{x}\)), whereas run 2 gives only the upper branch \(y = 2\sqrt{x}\).

Transform to cylindrical coordinates and compute \({\textcolor{gray}{\int}}_{\!\!\!0}^{2 \pi}{\textcolor{gray}{\int}}_{\!\!\!0}^{3}{\textcolor{gray}{\int}}_{\!\!\!0}^{4}r \textcolor{gray}{d}z \textcolor{gray}{d}r \textcolor{gray}{d}\theta\), where \(E\) is the solid bounded by \(z = 0\), \(z = 4\), and \(r = 3\).

Analysis: Run 1's answer \(144\pi\) and run 6's answer \(108\pi\) differ by a factor, representing distinct numerical values.

Find the volume of the solid bounded above by \(z = -x^{2}-y^{2}+4\) and below by \(z = x^{2}+y^{2}\).

Analysis: The three answers are distinct numeric values: run 1 gives \(\tfrac{8\pi}{3}\), run 2 gives \(2\pi\), and run 3 gives \(4\pi\).

Find the determinant of the 4x4 matrix \(A = \left[\begin{array}{cccc} 2 & 1 & 0 & 1 \\ 1 & 3 & 1 & 0 \\ 0 & 1 & 2 & 1 \\ 1 & 0 & 1 & 3 \end{array}\right]\).

Analysis: The answers give four distinct numerical values for the determinant: run 1 reports \(20\), run 4 reports \(28\), run 5 reports \(17\), and run 3 reports \(21\).

Find one valid singular value decomposition of the matrix \(\left[\begin{array}{cc} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{array}\right]\).

Analysis: The first group all give valid SVD triples \((U, \Sigma, V)\) with column-orthonormal \(U\) and \(V\) and singular values \(\sqrt{(91\pm\sqrt{8185})/2}\), differing only by sign flips of corresponding columns of \(U\) and \(V\); the second group has \(U\) columns that are not unit vectors under the stated normalization, so the triple is a different decomposition.

Solve \(\frac{d^{2}}{d x^{2}}y \! \left(x \right) = \frac{x^{2} \left(\frac{d}{d x}y \! \left(x \right)\right)^{2}-2 x y \! \left(x \right) \left(\frac{d}{d x}y \! \left(x \right)\right)+y \! \left(x \right)^{2}}{x^{2}}\).

Analysis: The answer by run 1 is given in terms of the exponential integral \(\mathrm{Ei}(\_C_1-\ln x)\), while the answer by run 6 is given as \(x\tan(c_1\ln x+c_2)\), and no reparametrization of the constants makes these two functional forms coincide.

Solve the polynomial equation \(x^{4}-2 x^{3}-3 x^{2}+4 x +4 = 0\) over the reals, finding all roots.

Analysis: Run 1 gives the distinct real roots \(\{-1, 2\}\) as a set, while run 4 lists \(-1, -1, 2, 2\) with each root repeated, indicating multiplicity two for both roots.

Compute a lexicographic Gröbner basis (with \(y <x\)) of the ideal \(\langle x^{2}+y^{2}-4, x y -1\rangle\) and use it to find all complex solutions of the system.

Analysis: The first group provides both the lexicographic Gröbner basis \(\{y^4-4y^2+1,\ x+y^3-4y\}\) and the four solutions, while runs 2, 3, 4, and 6 omit the Gröbner basis and give only the four solutions (which are equivalent to one another, since \(\sqrt{2\pm\sqrt{3}} = \tfrac{1}{2}(\sqrt{6}\pm\sqrt{2})\)).

Perform a two-tailed one-sample t-test at \(\alpha = 0.05\) for \(\mathit{H0}\) using the sample \([9, 10, 11, 12, 13]\) (assume normal population with unknown variance).

Analysis: The run 1 answer is \(\sqrt{2} \approx 1.414\), while run 2's answer is \(1 - \tfrac{4\sqrt{3}}{9} \approx 0.230\), which are numerically and algebraically distinct.

Find the number of distinct necklaces with 6 beads using 3 colors, where rotations are considered identical.

Analysis: The answers \(36\) and \(92\) are distinct integers, so run 2 and run 1 give numerically different counts of necklaces.