Gemini: on the 100-problem benchmark

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

Analysis: Run 4's answer \(2\zeta(3)\) is a constant involving Apéry's constant, while run 2's answer \(2 - \pi^2/6\) is a rational combination of \(\pi^2\); these are numerically and symbolically distinct values.

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

Analysis: Run 1 gives \(\pi^2/8\) while run 2 gives Catalan's constant \(G \approx 0.9159\), which are numerically and symbolically distinct (\(\pi^2/8 \approx 1.2337\)).

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

Analysis: The first group locates both critical points \((0,0)\) and \((1,1)\) and classifies them as a saddle and a local minimum, while the second group lists the points without any classification.

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: The first group provides both the maximum rate \(4\sqrt{2}\) and the direction (given as \(\langle 4,-4\rangle\) by run 3 and run 4, and as the unit vector \(\langle \sqrt{2}/2,-\sqrt{2}/2\rangle\) by run 2, which point the same way), while run 6 reports only the magnitude \(4\sqrt{2}\) with no direction.

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 two answers are different numerical values, \(-6\) and \(12\), which are not equal.

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

Analysis: The decomposition by run 5 has \(R_{22} \approx 0.8165\) while run 1 has \(R_{22} = \sqrt{6}/2 \approx 1.2247\), so the second columns of \(Q\) and \(R\) are not related by a uniform sign flip and the two factorizations are not the same.

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 a full SVD with \(U, \Sigma, V\) and singular values \(9.5255, 0.5143\); run 1 reports \(U, \Sigma, V\) but with a second singular value \(0.9635\) and a correspondingly different \(V\); run 4 returns only the two singular values \(9.5255, 0.5143\) without any \(U\) or \(V\).

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

Analysis: The first group expresses \(y=-u'/u\) (in various Bessel bases of \(u''+x^2u=0\)), while the remaining answer has the opposite overall sign, corresponding to \(+u'/u\).

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: run 1 and run 6 give an algebraic–logarithmic implicit solution \(4x^3-6x^2+12x-6y-3\ln((x+1)^4(2x^2-2y+1))=C\); run 2 has \(x/y+\tfrac{\sqrt\pi}{2}\mathrm{erf}(x)=C\); run 3 and run 4 have \(x/y+\tfrac{\sqrt\pi}{2}\mathrm{erf}(y)=C\); run 5's \(\int e^{-x^2}\,dy = y\,e^{-x^2}\) matches none of the others.

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 first group expresses the general solution in terms of the exponential integral \(\mathrm{Ei}\), while \(y=C_1 x\) is only the one-parameter linear solution and the last is a different polynomial–logarithmic form \(x^2(C_1-\ln|x|)+C_2 x\) without \(\mathrm{Ei}\).

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: All share the closed form \(c_n = 8/(\alpha_n^3 J_1(\alpha_n))\), but every answer reports a different set of 4-decimal numerical values for \(c_1, c_2, c_3\).

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

Analysis: run 3 gives both the full set of six roots and the upper-half-plane subset, run 4 gives only the full set of six roots (using \(2e^{i\theta}\) instead of \(\operatorname{polar}(2,\theta)\)), and run 6 gives only the upper-half-plane subset.

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: Run 1's coefficients for the regular part are doubled (\(-1/2, -z/4, -z^2/8, \ldots\)) compared to the others' \(-1/4, -z/8, -z^2/16, \ldots\).

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: Three distinct mathematical answers: runs 1 and 2 give both expansions as a tuple (the \(0<|z|<1\) and \(|z|>1\) Laurent series); runs 3, 4, and 6 give only the \(0<|z|<1\) expansion \(-1/z^2 - 1/z - 1 - z - z^2\); run 5 gives only the \(|z|>1\) expansion \(1/z^3 + 1/z^4 + 1/z^5 + 1/z^6 + 1/z^7\). The largest equivalent group has 3 members.

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

Analysis: The answers differ in sign: run 1 gives \(-4\) while run 6 gives \(4\).

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: run 1 gives both the discriminant \(-4p^3 - 27q^2\) and the repeated-root condition \(-4p^3 - 27q^2 = 0\); run 3 gives only the discriminant; run 4 gives only the condition equation.

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

Analysis: The first group all simplify to \(\sin(5x)+\sin(3x)-\sin(7x)-\sin(x)\), while the lone answer uses pure cosines \(\cos(x)-\cos(3x)-\cos(5x)+\cos(7x)\), which is a different function of \(x\).

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: run 2 gives only the residual sum of squares \(0.259\), while run 4 and run 6 also provide the fitted coefficients \(a=1.17\) and \(b=1.21\) along with the same residual sum of squares.

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: run 2 gives the regression line's coefficients as the triple \((a, b, R^2) = (2.2, 0.6, 0.6)\), while run 1 gives the line itself as the expression \(2.2 + 0.6x\) together with \(R^2 = 0.6\), omitting the intercept and slope as separate scalars.

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: Run 1 gives the numerical value \(0.2346\), run 2 gives \(0.2285\) (a distinct numerical value), and run 4 gives \(\sqrt{2}\approx 1.4142\), which is a different quantity (the t-statistic rather than a probability).

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

Analysis: Run 5 gave only the numerical value \((1+\sqrt{5})/4 \approx 0.809\), while run 1 gave the minimal polynomial \(8x^3-6x+1\) together with the numerical root \(0.766044\), which is the actual value of \(\cos(2\pi/9)\).