AI vs AI: disagreements on the 100-problem benchmark

Evaluate \(\underset{n \rightarrow \infty}{\mathrm{lim}}\frac{\overset{n}{\underset{k =1}{\sum}}\sqrt{1+\frac{k}{n}}}{n}\).

Analysis: deepseek's \(0.884488434\) is numerically about \(0.884\), while codex's \((4\sqrt{2}-2)/3\) is approximately \(1.219\).

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

Analysis: deepseek gives \(0\) while codex gives \(\frac{1}{120}\), which are distinct numerical values.

Find the inflection points, as ordered pairs, and intervals of concavity of \(f \! \left(x \right) = x^{4}-8 x^{2}+3\).

Analysis: The first group gives inflection points \((\pm 2/\sqrt{3}, -53/9)\) with concave up on \((-\infty,-2/\sqrt{3})\cup(2/\sqrt{3},\infty)\) and concave down on \((-2/\sqrt{3},2/\sqrt{3})\); deepseek lists three points including \((0,3)\) with \(y=-13/9\) and the opposite concavity assignment; mistral uses \(\pm(2/3)\sqrt{6}\) with \(y=-163/9\) plus \((0,3)\) and an alternating four-interval concavity pattern.

Apply 3 iterations of Gauss-Seidel to the system \(10 x_{1}-x_{2}+2 x_{3} = 6\), \(-x_{1}+11 x_{2}-x_{3} = 25\), \(2 x_{1}-x_{2}+10 x_{3} = -11\), starting from \([x_{1}, x_{2}, x_{3}] = [0, 0, 0]\). Give \([x_{1}, x_{2}, x_{3}]\) after iteration 3, each to 6 decimal places.

Analysis: The deepseek answer gives \([0.524286, 2.426667, -1.184524]\), mistral gives \([0.601652, 2.327052, -0.988682]\), and codex gives \([1.043338, 2.269542, -1.081713]\), three distinct numerical triples.

Assuming \(0<a\), compute the curvature \(\kappa \! \left(t \right)\) and torsion \(\tau \! \left(t \right)\) of the helix \(r \! \left(t \right) = \langle a \cos \! \left(t \right), a \sin \! \left(t \right), b t \rangle\).

Analysis: deepseek gives only \(\tau=b/(a^2+b^2)\) with no curvature; mistral gives \(\kappa=b^2/(a(a^2+b^2))\) with \(\tau=b/(a^2+b^2)\); codex gives \(\kappa=a/(a^2+b^2)\) with \(\tau=b/(a^2+b^2)\).

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

Analysis: The deepseek answer is \(0\), the mistral answer is \(9\), and the codex answer is \(7\).

Find the closed form of the infinite sum \(\overset{\infty}{\underset{n =1}{\sum}}\frac{H_{n}}{n^{2}}\), where \(H_{n} = \overset{n}{\underset{k =1}{\sum}}\frac{1}{k}\) is the n-th harmonic number. Express the result in terms of the Riemann zeta function.

Analysis: deepseek and codex give \(2\zeta(3)\), while mistral gives \(\zeta(3) + \frac{\pi^2 \ln 2}{12}\).

Evaluate \(\int_{0}^{1}\ln \! \left(x \right) \ln \! \left(1-x \right)d x\) in closed form.

Analysis: Deepseek gives \(-2 + \pi^2/6\), codex gives \(2 - \pi^2/6\) (the negative of deepseek), and mistral gives \(2\zeta(3) - 2\), which involves \(\zeta(3)\) rather than \(\pi^2\).

Find the closed form of the infinite sum \(\overset{\infty}{\underset{n =1}{\sum}}\frac{\cos \! \left(n \right)}{n^{2}}\) (the argument is in radians, not degrees).

Analysis: Mistral's expression evaluates (with \(\operatorname{csgn}(i e^{i}) = -1\)) to \(\pi^2/2 + \pi - 1/4\), whereas codex's expression is \(\pi^2/6 - \pi/2 + 1/4\).

Decompose into partial fractions and evaluate \(\int \frac{3 x^{2}+2 x +1}{\left(x -1\right) \left(x +2\right)^{2}}d x\).

Analysis: The codex/claude/grok group gives \(\frac{2}{3}\ln|x-1| + \frac{7}{3}\ln|x+2| + \frac{3}{x+2}\); deepseek gives \(-2\ln|x-1| + \ln|x+2| + \frac{3}{x+2}\); mistral gives \(\frac{1}{3}\ln(x-1) - \frac{11}{9}\ln(x+2) - \frac{11}{3(x+2)} + \frac{-3x-4}{3(x+2)^2}\).

Evaluate \(\int_{0}^{1}\frac{\arctan \! \left(x \right)}{x}d x\) in closed form.

Analysis: Codex and grok give Catalan's constant \(G\), while deepseek gives an arctan of a complex dilogarithmic expression.

Find the total area enclosed by both loops of the lemniscate \(r^{2} = 4 \cos \! \left(2 \theta \right)\).

Analysis: claude gives \(8\), mistral gives \(4\sqrt{2}\), and codex gives \(4\), three distinct numerical values.

Evaluate \(\int_{0}^{1}\frac{\ln \! \left(x^{2}+1\right)}{x^{2}+1}d x\) in closed form. Express the result using \(\pi\), \(\ln \! \left(2\right)\), and Catalan's constant \(G\).

Analysis: The codex and grok answers give \(\frac{\pi \ln 2}{2} - G\), while the deepseek answer gives \(\frac{\pi \ln 2}{4} - \frac{G}{2}\), which is exactly half of the other group's value.

Find the critical points of \(f \! \left(x , y\right) = x^{3}+y^{2}-3 x\) and classify them.

Analysis: The grok answer lists only the critical points \((1,0)\) and \((-1,0)\) without classifying them, while the others identify \((1,0)\) as a local minimum and \((-1,0)\) as a saddle point.

Locate and classify the critical points of the surface defined by \(f \! \left(x , y\right) = x^{3}+y^{3}-3 x y\).

Analysis: Gemini's answer omits the classification of each critical point as saddle or local minimum, while the others identify both \((0,0)\) as a saddle and \((1,1)\) as a local minimum.

Find the maximum rate of change of \(f \! \left(x , y\right) = x^{2} {\mathrm e}^{-y}\) at \([2, 0]\) and the direction in which it occurs.

Analysis: deepseek gives only the scalar \(\sqrt{17}\); gemini gives only the scalar \(4\sqrt{2}\); mistral gives \(4\sqrt{5}\) paired with direction \([4,-4]\); codex gives \(4\sqrt{2}\) paired with unit direction \([1/\sqrt{2},-1/\sqrt{2}]\).

Find the extreme values of \(f \! \left(x , y\right) = x^{2}+2 y^{2}\) subject to the constraint \(x +y = 3\).

Analysis: codex gives the minimum \(6\) and an unbounded maximum \(\infty\); deepseek gives \(-6\) and \(12\); mistral gives the interval \([3, 18]\); claude gives only \(6\).

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: The answers are \(36\pi\), \(108\pi\), \(\frac{81}{2}\pi\), and \(72\pi\) respectively, all numerically distinct.

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 deepseek, codex, and grok answers give \(4\pi\), while the mistral answer gives \(\pi\).

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: deepseek says \(20\), claude says \(21\), mistral says \(10\), and codex says \(12\).

Find the reduced row echelon form of \(A = \left[\begin{array}{cccc} 1 & 2 & -1 & 3 \\ 2 & 4 & 1 & -1 \\ 3 & 7 & 0 & 2 \end{array}\right]\).

Analysis: Codex, claude, gemini and grok all give \(\begin{pmatrix}1&0&0&2/3\\0&1&0&0\\0&0&1&-7/3\end{pmatrix}\); deepseek gives \(\begin{pmatrix}1&0&0&13\\0&1&0&-7\\0&0&1&4\end{pmatrix}\); mistral gives \(\begin{pmatrix}1&0&0&13/3\\0&1&0&-7/3\\0&0&1&-1\end{pmatrix}\).

Find the characteristic polynomial and eigenvalues of the companion matrix \(A = \left[\begin{array}{ccc} 0 & 0 & 2 \\ 1 & 0 & -1 \\ 0 & 1 & 2 \end{array}\right]\).

Analysis: The gemini answer gives the negative of the characteristic polynomial \(-\lambda^3+2\lambda^2-\lambda+2\) (using the \(\det(A-\lambda I)\) convention), while the others give the monic form \(\lambda^3-2\lambda^2+\lambda-2\); eigenvalues agree across all answers.

Find one valid QR decomposition of the matrix \(\left[\begin{array}{cc} 1 & 1 \\ 1 & 0 \\ 0 & 1 \end{array}\right]\).

Analysis: Mistral's answer is a pair of scalars while codex's answer is a pair of matrices \(Q\) and \(R\) forming a QR decomposition of \(\begin{pmatrix}1&1\\1&0\\0&1\end{pmatrix}\).

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 gives the thin/reduced SVD with \(U\in\mathbb{R}^{3\times 2}\) and \(\Sigma\in\mathbb{R}^{2\times 2}\), while the second gives the full SVD with \(U\in\mathbb{R}^{3\times 3}\) and \(\Sigma\in\mathbb{R}^{3\times 2}\) (including the extra orthonormal column \([1,-2,1]/\sqrt{6}\) in \(U\) and an extra zero row in \(\Sigma\)).

Diagonalize \(A = \left[\begin{array}{cc} 4 & 1 \\ 2 & 3 \end{array}\right]\) as \(A = D\), giving \(P\) and \(D\) explicitly.

Analysis: Codex, deepseek, claude, gemini and grok give \(D = \operatorname{diag}(5, 2)\) with \(P\) columns spanning the same eigenspaces (second column \([1,-2]\) or its scalar multiple \([-1/2, 1]\)); mistral's \(P\) uses \([1, 2]\) as the second column, which lies in a different direction from the \([1,-2]\) eigenvector.

Find the minimum-norm least squares solution to \(A x = b\) for \(A = \left[\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}\right]\) and \(b = \left[\begin{array}{c} 1 \\ 1 \end{array}\right]\).

Analysis: Claude, gemini, and codex give \((-1/2, 0, 1/2)\); deepseek gives \((-1/3, -1/6, 0)\); mistral gives \((-1/3, 0, 1/3)\).

Solve the Riccati equation \(\frac{d}{d x}y \! \left(x \right) = 1+x^{2}-2 x y \! \left(x \right)+y \! \left(x \right)^{2}\).

Analysis: codex and claude both give \(x - 1/(x+C)\), equivalent to \(x + 1/(C-x)\); deepseek has the opposite sign \(x + 1/(x+C)\), a different one-parameter family; mistral expresses the answer through Bessel \(I\) and \(K\) functions, a non-elementary form distinct from the rational ones.

Solve the Riccati equation \(\frac{d}{d x}y \! \left(x \right) = y \! \left(x \right)^{2}+x^{2}\) in closed form.

Analysis: The codex, gemini, and grok answers are the same general solution written as \(-u'/u\) with \(u=\sqrt{x}\,(C_1 J_{1/4}(x^2/2)+\text{second solution})\), expressed in equivalent Bessel bases; the deepseek answer is not a single ratio (operator precedence leaves it as a sum plus a fraction) so it does not represent that solution; the mistral answer drops the arbitrary constant and adds an extra \(-x\) term, giving a different expression.

Solve \(\frac{d}{d x}y \! \left(x \right) = \frac{x \left(x^{2}-x -1-2 x^{2} y \! \left(x \right)+2 x^{4}\right)}{\left(x^{2}-y \! \left(x \right)\right) \left(x +1\right)}\), expressing the implicit general solution.

Analysis: The codex/claude/gemini/grok group gives \(-2y + \frac{4}{3}x^3 - 2x^2 + 4x - \ln|(2x^2-2y+1)(x+1)^4| = C\); deepseek gives a polynomial form with \(y^2\), \(xy\), \(x^2 y\) terms plus \(\ln(x+1) + \ln(x^2-y)\); mistral gives \(-\ln|(x^2-y)(x+1)| - x^2 + \ln(x) = C\), missing the \(2x^2-2y+1\) and cubic terms.

Solve Bessel's equation \(x^{2} \left(\frac{d^{2}}{d x^{2}}y \! \left(x \right)\right)+x \left(\frac{d}{d x}y \! \left(x \right)\right)+\left(x^{2}-1\right) y \! \left(x \right) = 0\).

Analysis: deepseek and codex give the general solution with two arbitrary constants \(C_1 J_1(x) + C_2 Y_1(x)\), while mistral has only one arbitrary constant, leaving the \(J_1(x)\) term without a free coefficient.

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 codex/grok pair expresses \(y\) using the exponential integral \(\operatorname{Ei}\) of \(\ln x\), deepseek uses \(x\exp(C/x)\), claude uses \(\tan(c_1\ln x+c_2)\), gemini uses \(x^2(C_1-\ln|x|)+C_2 x\), and mistral uses \(\sin(\ln x)\) and \(\cos(\ln x)\) — five algebraically distinct closed forms.

Determine the series solution for \(\frac{d^{2}}{d x^{2}}y \! \left(x \right)+x y \! \left(x \right) = 0\) (Airy's equation) around \(x = 0\) up to order 5.

Analysis: The gemini/claude group gives both independent solutions truncated through \(x^4\); deepseek gives only one branch and extends it to \(x^6\); mistral gives both branches but extends them to \(x^6\) and \(x^7\).

For the Fourier-Bessel expansion \(f \! \left(r \right) = -r^{2}+1 = \overset{\infty}{\underset{n =1}{\sum}}c_{n} J_{0}\! \left(\alpha_{n} r \right)\) on \([0, 1]\), where \(\alpha_{n}\) is the n-th positive zero of \(J_{0}\! \left(x \right)\), give the closed form for \(c_{n}\) and the numeric values of \(c_{1}\), \(c_{2}\), \(c_{3}\) to 4 decimal places.

Analysis: claude and codex give \(c_n = 8/(\alpha_n^3 J_1(\alpha_n))\) with values \(1.1080, -0.1398, 0.0455\); gemini has the same closed form but \(c_1 = 1.1061\) and \(c_2 = -0.1396\); mistral has the same closed form but numerics \(4.2359, -1.0409, 0.4575\); deepseek has a different closed form \(2/(\alpha_n^2 J_1(\alpha_n))\) with values \(0.2161, -0.0648, 0.0332\).

Find the harmonic conjugate of \(u \! \left(x , y\right) = x^{2}-y^{2}\).

Analysis: deepseek's answer \(x^2 + 2xy - y^2\) contains the additional term \(x^2 - y^2\) that codex's answer \(2xy\) does not.

Find all 6th roots of \(-64\) in polar form and identify which roots lie in the upper half-plane.

Analysis: The first group lists all six roots and identifies the three in the upper half-plane; the second group lists only the three upper-half-plane roots without the full set of six; the third group gives values with modulus \(2^{4/3}\) instead of \(2\).

Evaluate \(\int \frac{{\mathrm e}^{z}}{\left(z -1\right)^{3}}d z\) over the contour \({| z |} = 3\).

Analysis: The codex/grok group gives \(\pi i e\), mistral gives half of that \(\frac{1}{2}\pi i e\), and deepseek gives \(-2\pi i e\).

Find the Laurent series expansion of \(f \! \left(z \right) = \frac{1}{z \left(z -2\right)}\) about \(z = 0\) for \(0<{| z |}<2\).

Analysis: The first group gives \(-\frac{1}{2z} - \sum_{n=0}^{\infty} \frac{z^n}{2^{n+2}}\), while mistral has \(+\frac{1}{2z}\) and alternating signs in the power-series part.

Find the residue of \(f \! \left(z \right) = \frac{\sin \! \left(z \right)}{z^{4}}\) at \(z = 0\).

Analysis: deepseek gives \(0\) while codex gives \(-\frac{1}{6}\), which are distinct numerical values.

Find the Laurent series of \(f \! \left(z \right) = \frac{1}{z^{2} \left(z -1\right)}\) around \(z = 0\) in both regions: (a) \(0<{| z |}<1\) and (b) \(1<{| z |}\), each with 5 non-zero terms.

Analysis: Group one provides both Laurent series for regions (a) \(0<|z|<1\) and (b) \(|z|>1\), while group two provides only the series for region (b).

Find the residue of \(f \! \left(z \right) = \frac{{\mathrm e}^{z}}{\left(z -i \pi \right)^{2}}\) at \(z = i \pi\).

Analysis: codex gives \(-1\), mistral gives \(\frac{1}{2}e^{i\pi} = -\frac{1}{2}\), and deepseek gives \(0\).

Evaluate \(\int_{0}^{2 \pi}\frac{1}{2+\cos \! \left(\theta \right)}d \theta\).

Analysis: The deepseek answer \(0.5773502691896257 \approx 1/\sqrt{3}\) differs from the codex answer \(2\pi/\sqrt{3} \approx 3.6276\) by a factor of \(2\pi\).

Find the resultant of \(p = x^{3}-2\) and \(q = x^{2}-2\) with respect to \(x\).

Analysis: Gemini gives the scalar \(4\), codex gives \(-4\) (opposite sign), and mistral gives the polynomial \(x^5 - 2x^3 - 2x^2 + 4\) rather than a scalar resultant.

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

Analysis: The mistral answer lists four RootOf placeholders for all roots (including complex ones) of \(x^4-2x^3-3x^2+4x+4=0\), while codex gives only the two values \(\{-1, 2\}\), so they denote different root sets.

Solve the nonlinear system \(x^{3}+y^{3} = 35\), \(x +y = 5\) over the reals.

Analysis: The deepseek answer gives a single pair \([5, 0]\) which does not satisfy \(x^3 + y^3 = 35\), while gemini and grok both give the two solutions \((x, y) = (2, 3)\) and \((3, 2)\).

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 codex/claude/gemini group gives the four solution pairs \((\pm\sqrt{2\pm\sqrt{3}},\,\pm\sqrt{2\mp\sqrt{3}})\) (and codex's basis \([x+y^3-4y,\,y^4-4y^2+1]\) matches); deepseek gives \((\pm\sqrt{3},\,\pm\sqrt{3})\); mistral's basis contains the cubic \(y^3-4y+1\) instead of a quartic in \(y\); grok's basis has \(x-y^3+4y\), opposite in sign to codex's \(x+y^3-4y\).

Determine the Galois group of \(x^{4}-2\) over \(Q\).

Analysis: codex, claude, and grok give the dihedral group of order 8 (\(D_4\)); deepseek gives the cyclic group \(C_4\); mistral gives the group generated by \((2,3,4)\) and \((1,2)\), which is \(S_4\).

Find the discriminant of \(f \! \left(x \right) = x^{3}+p x +q\) in terms of \(p\) and \(q\), and state the condition on \([p, q]\) for \(f\) to have a repeated root.

Analysis: deepseek and grok give the discriminant \(-4p^3 - 27q^2\) (with deepseek additionally stating the repeated-root condition as \(=0\)), while claude states only the condition \(-4p^3 - 27q^2 = 0\) without separately giving the discriminant expression.

Evaluate \(\int_{0}^{\frac{\pi}{2}}\ln \! \left(\sin \! \left(x \right)\right)^{2}d x\) in closed form.

Analysis: deepseek's answer has a negative sign on the \(\pi^3/24\) term, while codex and grok give \(\pi^3/24 + (\pi/2)\ln(2)^2\).

Express \(4 \sin \! \left(x \right) \sin \! \left(2 x \right) \sin \! \left(4 x \right)\) as a sum of cosines.

Analysis: The two cosine expressions differ by a factor of \(16\) in their coefficients, while the third is a sum of sines \(-\sin(7x)+\sin(5x)+\sin(3x)-\sin(x)\) which is numerically distinct from both cosine forms.

Find the Fourier series expansion (first four nonzero terms) of the square wave \(f \! \left(x \right) = \mathrm{signum}\! \left(\sin \! \left(x \right)\right)\) on \([-\pi, \pi]\).

Analysis: The deepseek answer lists only the Fourier coefficients \(0, 4/\pi, 0, 4/(3\pi), \ldots\) without the \(\sin(nx)\) basis functions, while the others give the full series \(\frac{4}{\pi}\sum \frac{\sin((2k-1)x)}{2k-1}\).

Find the Stirling number of the second kind \(S \! \left(6, 3\right)\).

Analysis: The deepseek answer is \(20\), the mistral answer is \(25\), and the codex answer is \(90\), three distinct numerical values.

Find the number of binary strings of length 8 with no consecutive 1's.

Analysis: deepseek's answer is \(34\) while codex's answer is \(55\), which are distinct integers.

Find the number of ways to partition the set \(\{1, 2, 3, 4\}\) into non-empty subsets.

Analysis: The number of set partitions of {1,2,3,4} is the Bell number \(B_4 = 15\), while deepseek's \(5\) corresponds to \(B_3\) or the Stirling number \(S(4,2)\), so the two values are not equivalent.

Find the number of labeled trees on 6 vertices.

Analysis: Deepseek's 720 equals \(6!\) while codex's 1296 equals \(6^4\), which is the value given by Cayley's formula \(n^{n-2}\) for labeled trees on 6 vertices.

Find the number of spanning trees in the complete bipartite graph \(K_{2,3}\).

Analysis: deepseek gives 10 while codex gives 12 as the count of spanning trees in \(K_{2,3}\).

Find the number of ways to partition a 6-element set into 3 unordered subsets of size 2.

Analysis: deepseek gives 10 while codex gives 15 for the count of partitions of a 6-element set into 3 unordered subsets of size 2.

For data \(X = [1, 2, 3, 4, 5]\) and \(Y = [ 2.1, 3.9, 4.8, 6.2, 7.0]\), fit a least squares regression line \(y = b x +a\) and compute the residual sum of squares.

Analysis: Codex and gemini give the full triple \([1.17, 1.21, 0.259]\) (slope, intercept, residual sum of squares); claude and grok give only the residual sum of squares \(259/1000 = 0.259\); mistral gives \([a = 0.66,\ b = 1.28,\ \mathit{rss} = 0.308]\); deepseek gives \(0.028\).

Find the range and interquartile range of \([5, 12, 18, 23, 28, 35, 41]\) using Tukey's hinges, with the median excluded from the lower and upper halves.

Analysis: deepseek listed the original data instead of giving range and IQR; gemini and claude give range \(36\) and IQR \(23\); mistral gives \(29\) and \(18\).

For data \(\{1, 2, 3, 4, 5\}\), compute the least squares regression line \(y = b x +a\) and the coefficient of determination \(R^{2}\).

Analysis: The first group gives slope \(3/5\), intercept \(11/5\), and \(R^2 = 3/5\); deepseek reports \(0.2,\ 0.9,\ 0.45\); mistral reports slope \(0.7\), intercept \(1.1\), and \(R^2 = 0.7\).

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: Codex, gemini and grok give a p-value \(\approx 0.2302\) (grok's symbolic form \(1 - 4\sqrt{3}/9\) evaluates to the same value); claude gives the t-statistic \(\sqrt{2} \approx 1.414\); mistral reports \(t = -0.7071\) with \(p \approx 0.519\); deepseek returns \(\mathrm{NaN}\).

For \(\mathit{X\,~\,Beta(\alpha=2,\,\beta=3)\,}\), find \(E_{X}\), \(\mathit{Var}_{X}\), and the mode.

Analysis: Codex gives \(E[X]=2/5\), \(\text{Var}[X]=1/25\), mode \(=1/3\); deepseek gives \(E[X]=2/5\), \(\text{Var}[X]=1/25\), but mode \(=1/4\); mistral gives \(E[X]=8/5\), \(\text{Var}[X]=4/125\), mode \(=1/3\).

Find median and sample standard deviation (denominator \(n -1\)) of \([3, 7, 8, 12, 15]\).

Analysis: The answers differ in both the median and the standard deviation: mistral gives median \(8\) with SD \(\frac{\sqrt{65}}{2}\), claude gives median \(8\) with SD \(\sqrt{43/2}\), and deepseek gives median \(9\) with SD \(\approx 4.7434\).

Find the linear regression equation \(y = a x +b\) that best fits the points \([1, 2]\), \([2, 4]\), \([3, 5]\).

Analysis: The answers differ in slope and intercept: claude gives \(a=3/2, b=2/3\); mistral gives \(a=3/2, b=5/6\); deepseek gives \(a=1, b=4/3\).

Evaluate \(\int_{0}^{1}\frac{\ln \! \left(x \right)^{2}}{x^{2}+1}d x\) in closed form.

Analysis: deepseek gives \(-\pi^3/16\), mistral gives \(\pi^3/16 - \pi(\ln 2)^2/8 + i\,\text{Catalan}/2\), and codex gives \(\pi^3/16\).

Find the equation of the osculating circle to the curve \(y = x^{2}\) at the point \([1, 1]\).

Analysis: The answers give three different circles: codex's has center \((-4, 7/2)\) and radius² \(125/4\), deepseek's has center \((0, 5/2)\) and radius² \(5/4\), and mistral's has center \((-1/2, 5/4)\) and radius² \(13/16\).

Evaluate \(\int_{0}^{1}\frac{\ln \! \left(1-x \right)^{2}}{x}d x\) in closed form. Express the result using \(\zeta \! \left(3\right)\).

Analysis: The claude and codex answers give \(2\zeta(3)\), while deepseek gives \(-2\zeta(3)\), differing in sign.

Evaluate \(\int_{0}^{\infty}\frac{\ln \! \left(x \right)}{\left(x^{2}+1\right)^{2}}d x\) in closed form.

Analysis: deepseek gives \(0\), mistral gives \(-\pi^2/16\), and codex gives \(-\pi/4\), three distinct numerical values.

Find the Jacobian determinant of the transformation \(u = x^{2}-y^{2}\), \(v = 2 x y\).

Analysis: The deepseek answer is \(-4(x^2+y^2)\) while the codex answer is \(4x^2+4y^2\); they differ by a sign.

Evaluate \(\int F d C\) where \(F = \langle z , x , y\rangle\) and \(C\) is the circle \(x^{2}+y^{2} = 1\), \(z = 1\), oriented counterclockwise when viewed from above.

Analysis: Deepseek's answer is \(0\) while codex's answer is \(\pi\), which are distinct numerical values.

Project the vector \(\langle 2, 3, 4\rangle\) onto the subspace of \(R^{3}\) spanned by \(\langle 1, 1, 1\rangle\) and \(\langle 1, 0, 0\rangle\).

Analysis: The deepseek answer \([2,2,2]\) and the mistral answer \([14/3, 7/3, 7/3]\) each differ from the claude/gemini/codex result \([2, 7/2, 7/2]\) by different projection vectors.

Solve the Riccati equation \(\frac{d}{d x}y \! \left(x \right) = y \! \left(x \right)^{2}+2 x y \! \left(x \right)+x^{2}-1\).

Analysis: The 2,3,5 group gives \(y = -x - \frac{1}{x+C}\); deepseek gives \(y = -x + \frac{1}{x+C}\) (opposite sign on the reciprocal term); mistral gives \(y = -x + \tan(x+C)\) (a tangent rather than a rational form).

Solve Laplace's equation \(\Delta u \! \left(r , \theta \right) = 0\) on the annulus \(1<r <2\) in polar coordinates, with boundary conditions \(u \! \left(1, \theta \right) = \cos \! \left(2 \theta \right)\) and \(u \! \left(2, \theta \right) = 0\). Give the explicit harmonic solution \(u \! \left(r , \theta \right)\).

Analysis: The three expressions evaluate to different functions of \(r\): deepseek gives \(\frac{4}{3}(1-r^{-2})\cos 2\theta\), mistral gives \(\frac{4}{15}(r^{2}-4r^{-2})\cos 2\theta\), and codex gives \(\frac{1}{15}(16r^{-2}-r^{2})\cos 2\theta\), which are pairwise inequivalent.

Find the residue of \(f \! \left(z \right) = {\mathrm e}^{\frac{1}{z}}\) at \(z = 0\).

Analysis: codex gives the residue as \(1\), deepseek gives \(0\), and mistral states the residue is undefined.

Evaluate \(\int_{0}^{2 \pi}\frac{\cos \! \left(\theta \right)}{5+4 \cos \! \left(\theta \right)}d \theta\).

Analysis: The four answers are four distinct numerical values: \(0\), \(2\pi/3\), \(\pi/2 - 3\pi\sqrt{5}/4\), and \(-\pi/3\).

Factor the polynomial \(x^{8}+x^{4}+1\) completely over the complex numbers.

Analysis: The first group's factorizations all multiply to \(x^8 + x^4 + 1\), while mistral's product contains four extra quadratic factors yielding a polynomial of degree 16 rather than 8.

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

Analysis: The deepseek answer \(92\) and codex answer \(130\) are distinct integer values.

Find the closed form of the infinite sum \(\overset{\infty}{\underset{n =1}{\sum}}\frac{H_{n}}{n^{3}}\), where \(H_{n} = \overset{n}{\underset{k =1}{\sum}}\frac{1}{k}\) is the n-th harmonic number.

Analysis: The deepseek answer is \(\frac{1}{4}\pi^2\zeta(3)-\frac{1}{2}\zeta(5)\), the mistral answer involves \(\operatorname{Li}_4(1/2)\) and \(\ln(2)\) terms, and the codex answer is \(\pi^4/72\) — three numerically distinct values.

Derive the minimal polynomial of \(\cos \! \left(\frac{2 \pi}{9}\right)\) over \(Q\) and give its numerical value to 6 decimal places.

Analysis: Codex, claude, gemini and mistral give both the minimal polynomial \(8x^3 - 6x + 1\) and the numerical value \(0.766044\); grok gives only the numerical value \(0.766044\) without the polynomial; deepseek gives only \(-0.173648\) (a different root of the same polynomial).

Solve the wave equation \(\frac{\partial^{2}}{\partial t^{2}}u \! \left(x , t\right) = c^{2} \left(\frac{\partial^{2}}{\partial x^{2}}u \! \left(x , t\right)\right)\) on \(0<x <\pi\) with \(0<c\), with boundary conditions \(u \! \left(0, t\right) = u \! \left(\pi , t\right) = 0\), initial displacement \(u \! \left(x , 0\right) = x \left(\pi -x \right)\), and initial velocity \(\left(\frac{\partial}{\partial t}u \! \left(x , t\right)\right)\! \left(x , 0\right) = 0\). Express the solution \(u \! \left(x , t\right)\) as a Fourier sine series with the coefficients given in closed form.

Analysis: The mistral answer has nonzero coefficients \(-16/(\pi n^3)\) for odd \(n\), while codex and grok both give \(+8/(\pi n^3)\) for odd \(n\) (zero for even \(n\)).